N^ 


THE  GROUNDWORK  OF 
PRACTICAL  NAVAL  GUNNERY 


THE   GROUNDWORK  OF 
PRACTICAL  NAVAL  GUNNERY 


A  Study  of  the  Principles  and  Practice  of  Exterior 
Ballistics,  as  Applied  to  Naval  Gunnery 

And  of  the 

Computation  and  Use  of  Ballistic  and  Range  Tables 


By  PHILIP  R.  ALGER 
Professor  of  Mathematics,  U.  S.  Navy 

Revised  and  Extended  to  Include  the  Formulas  and  Methods  of 

Colonel  James  M.  Ingalls,  U.  S.  Army 

By  the  Officers  on  Duty  in  the  Department  of  Ordnance  and  Gunnery 

United  States  Naval  Academy 

1914-15 


ANMAPOLIS,    MD. 
THE    UNITED   STATES    NAVAL   INSTITUTE 


1915 


^^ 


COPTRIGHT,    1915,    BY 

P.  C.  ALLEN 
Sec.  and  Tkeas.  U.  S.  Naval  Institute 


c^ 


Column 

9 

Column 

3, 

Column 

4. 

Column 

5. 

Column 

6. 

Column 

8. 

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10. 

Column 

11. 

Column 

12. 

PREFACE. 

DEPAETMENT  OF  OKDNANCE  AND  GUNNERY. 

U.  S.  Naval  Academy,  Annapolis,  Md.,  January  1,  1015. 

1.  Experience  with  the  course  of  instruction  of  midshipmen  in  the  science  of 
exterior  ballistics  seemed  to  indicate  that  some  modification  of  that  course  was  neces- 
sar}',  especially  in  view  of  the  fact  that  the  Bureau  of  Ordnance  in  its  work  uses  the 
formulcB  and  methods  devised  by  Colonel  James  ^I.  Ingalls,  U.  S.  Army,  in  the  place 
of  those  given  by  Professor  Philip  K.  Alger,  U.  S.  Navy,  in  the  text  book  prepared 
by  him  and  used  at  the  Naval  Academy  up  to  this  date,  for  the  computation  of  the 
data  contained  in  the  following  columns  of  the  range  tables  prepared  by  the  Bureau 
of  Ordnance  and  officially  issued  to  the  service: 

Angle  of  departure. 

Angle  of  fall. 

Time  of  flight. 

Striking  velocity. 

Drift. 

Maximum  ordinate. 

Change  of  range  for  variation  of  ±  50  foot-seconds  initial  velocity. 

Change  of  range  for  variation  of  ±  Aw  in  weight  of  projectile. 

Change  of  range  for  variation  of  density  of  air  of  ±  10  per  cent. 

2.  Exterior  ballistics  is  taught  at  the  Naval  Academy  in  order  that  the  midship- 
men, when  they  graduate  and  become  officers,  may  be  as  familiar  with  the  range 
tables,  with  the  data  contained  in  them,  and  with  the  methods  by  which  this  data  is 
obtained  and  used ;  in  other  words,  with  the  use  of  the  information  which  is  furnished 
to  aid  them  in  using  the  gams  successfully;  as  the  time  available  for  this  instruction 
will  permit.  This  being  the  case,  everything  has  been  omitted  that  does  not  bear 
directly  upon  the  point  assumed  to  be  at  issue,  and  no  formulae  have  been  retained 
that  are  not  of  use  in  connection  with  the  practice  of  naval  gunnery  and  the  use  of  the 
range  tables  and  of  the  gun  afloat,  or  in  the  computation  of  the  data  contained  in 
the  range  tables;  except  that  certain  preliminary  formulae  and  discussions  have 
necessarily  been  retained  as  vitally  requisite  for  a  thorough  understanding  of  the 
later  and  mor^  practical  parts  of  the  work.  An  effort  has  been  made  to  reduce  the 
mathematical  investigations  to  the  lowest  possible  minimum  consistent  with  a  clear 
understanding  of  the  practical  portions  of  the  work,  in  accordance  with  the  views 
officially  expressed  by  the  Navy  Department,  by  the  Superintendent  of  the  Naval 
Academy,  and  by  the  Academic  Board ;  but  the  subject  is  one  that  is  almost  purely 
mathematical,  and  which  requires  considerable  preliminary  mathematical  work  (as  is 
the  case  with  the  science  of  navigation)  before  the  practical  features  can  be  properly 
understood.  The  preliminary  discussions  of  the  trajectory  have  therefore  been 
retained,  but  have  been  restricted  as  much  as  possible ;  and,  throughout,  the  work  of 
revision  has  been  carried  out  with  the  sole  object  in  view  of  giving  to  the  midshipmen, 
in  the  shortest  possible  time,  a  thorough  practical  knowledge  of  the  underlying 
principles  of  naval  gunnerv^  and  of  the  computation  and  use  of  the  range  tables. 

3.  In  brief,  the  advantages  sought  by  this  revision  of  the  text  book  previously 
in  use  are: 


339502 


6     '  '•  PREFACE 

(a)  An  arrangement  that  would  appear  more  logical  and  consecutive  to  a  mid- 
shipman taking  up  the  study  of  the  subject  for  the  first  time. 

(b)  A  more  clear  distinction  between  the  methods  and  formula?  that  are  purely 
educational  and  those  that  are  actually  used  in  practice. 

(c)  The  rendering  more  easily  understood  of  quite  a  number  of  points  in  the 
older  text  book  that  seemed  in  the  past  to  give  great  trouble  to  the  midshipmen  in 
their  study  of  the  subject. 

(d)  The  modification  of  the  problems  given  in  the  text  book  to  make  them  apply 
to  modern  United  States  Naval  Ordnance.  The  problems  in  the  older  text  book  dealt 
largely  with  foreign  ordnance,  and  exclusively  with  guns,  projectiles,  velocities,  etc., 
that  are  now  obsolete,  or  nearly  so ;  and,  while  many  of  the  older  problems  have  been 
retained  as  valuable  examples  of  principles,  a  large  number  of  problems  dealing  with 
present-day  conditions  and  ordnance  have  been  added. 

(e)  An  effort  has  been  made  to  give  a  complete  discussion  of  the  practical  use 
of  the  range  tables,  a  subject  but  lightly  touched  upon  in  the  older  text  book.  In 
order  to  accomplish  this  a  large  number  of  officers,  in  the  Atlantic  fleet  and  elsewhere, 
were  requested  to  contribute  such  knowledge  as  they  might  have  on  this  subject,  and 
the  matter  received  from  them  has  been  incorporated  in  the  chapter  on  this  subject. 
The  discussion  of  this  point  should  therefore  include  all  the  most  up-to-date  practice 
in  the  use  of  the  range  tables. 

4.  In  preparing  this  revision  for  the  purpose  indicated  in  the  preceding  para- 
graphs, the  logical  treatment  of  the  subject  seemed  to  indicate  its  division  under  two 
general  heads,  as  follows : 

(A)  The  treatment  of  the  trajectory  as  a  plane  curve;  which,  in  turn,  logically 
subdivides  itself  under  two  sub-heads,  as  follows : 

(a)  General  definitions,  etc.;  the  trajectory  in  vacuum;  the  resistance  of  the  air 
and  the  retardation  due  thereto ;  the  ballistic  coefficient  in  its  fullest  form ;  the  equa- 
tion to  the  trajectory  in  air  under  certain  specified  and  limited  conditions ;  and  the 
approximate  determination  of  the  elements  of  the  trajectory  by  the  use  of  the  above 
special  equation.  In  other  words,  the  features  that  are  of  educational  rather  than  of 
practical  value,  but  which  are  necessary  to  an  understanding  of  the  practical  methods 
that  are  to  follow. 

(b)  The  more  exact  and  practical  theories  and  formulae;  that  is,  the  ones  that 
are  generally  used  in  practical  work.  This  subdivision  is  not  a  rigid  one,  as  it  will  be 
seen  that  some  of  the  approximate  formulae  and  methods  are  sufficiently  accurate  to 
permit  of  their  use  in  practice,  and  they  are  so  used ;  but  the  general  statement  of  the 
subdivision  may  be  accepted  as  logical,  with  this  one  reservation. 

(B)  A  consideration  of  the  variation  of  the  actual  trajectory  from  a  plane  curve, 
which  treats  of  the  influence  upon  the  motion  of  the  projectile  of  drift  and  wind,  and 
of  the  effect  upon  the  fall  of  the  projectile  relative  to  the  target  of  motion  of  the  gun 
and  target.  That  is,  having  treated  under  the  first  division  those  computations 
that  are  not  materially  affected  by  the  variations  of  the  trajectory  from  a  plane  curve ; 
in  the  second  division  we  treat  of  the  effect  of  such  deviations  upon  accuracy  of  fire. 
In  other  words,  there  are  here  to  be  discussed  the  steps  taken  to  overcome  the  inaccu- 
racies in  fire  caused  by  the  variation  of  the  trajectory  from  a  plane  curve,  in  order  to 
hit  a  moving  target  with  a  shot  fired  from  a  gun  mounted  on  a  moving  platform, 
when  there  is  a  wind  blowing. 

5.  Following  these  natural  and  logical  divisions  of  the  subject  comes  a'  full 
discussion  of  the  range  tables,  column  by  column,  and  of  the  methods  of  computing 
the  data  contained  in  them,  and  of  using  this  data  after  it  has  been  computed  and 
tabulated. 


PEEFACE  7 

6.  Following  this  comes  a  consideration  of  the  processes  necessary  to  ensure  that 
the  guns  of  a  ship  shall  be  so  sighted  that  the  shot  in  a  properly  aimed  salvo  will  fall 
well  bunched. 

7.  Following  this  again,  comes  naturally  a  consideration  of  the  inherent  errors 
of  guns,  and  of  the  accuracy  and  prol)al)ility  of  fire. 

8.  In  accordance  with  the  preceding  statement  of  the  natural  and  logical  order 
in  which  the  subject  should  be  treated,  this  revised  text  book  is  therefore  divided  into 
six  parts,  as  follows: 

PAET  I. 

Chapters  1  to  5  Inclusive, 

general  and  approximate  deductions, 

Preliminary  definitions  and  discussion.  The  trajectory  in  vacuum.  The  resist- 
ance of  the  atmosphere  to  the  passage  of  a  projectile  through  it,  and  the  retardation 
in  the  motion  of  the  projectile  resulting  from  this  resistance.  The  ballistic  coefiBcient. 
The  equation  to  the  trajectory  in  air  when  Mayevski's  exponent  is  taken  as  having  a 
value  of  2;  and  a  comparison  between  this  equation  and  that  to  the  corresponding 
trajectory  in  vacuum.  The  derivation  from  this  special  equation  to  the  trajector}'  in 
air  of  certain  expressions  for  the  approximate  determination  of  the  values  of  the 
elements  of  the  trajectory. 

PAET  II. 
Chapters  6  to  12  Inclusive. 

PRACTICAL  methods. 

The  computation  and  use  of  the  ballistic  tables,  and  of  the  time  and  space 
integrals.  The  differential  equations  giving  the  relations  between  the  several  ele- 
ments of  the  trajectory  in  air.  Siacci's  method.  The  time,  space,  altitude  and 
inclination  functions,  and  their  computation  and  use.  The  ballistic  formulae. 
Ballistic  problems. 

PAET  III. 
Chapters  13  to  15  Inclusive. 

THE  variation  OF  THE  TRAJECTORY  FROM  A  PLANE  CURVE. 

The  variation  of  the  trajectory  from  a  plane  curve;  the  forces  that  cause  this 
variation ;  and  the  consideration  that  it  is  necessary  to  give  to  it  in  the  computations 
of  exterior  ballistics  and  in  the  use  of  the  gun.  Drift  and  the  theor}^  of  sights.  The 
effect  upon  the  travel  of  the  projectile  of  wind  and  of  motion  of  the  gun;  and  the 
effect  of  motion  of  the  target  upon  the  fall  of  the  projectile  relative  to  the  target. 

PAET  IV. 
Chapters  16  to  17  Inclusive. 

RANGE  tables  ;  THEIR  COMPUTATION  AND  USE. 

The  computation  of  the  data  contained  in  the  range  tables  and  the  practical 
methods  of  using  this  data. 


8  PREFACE 

PART  V. 

Chapters  18  to  19  Inclusive, 
the  calibration  of  single  guns  and  of  a  ship's  battery. 

The  determination  of  the  error  of  the  setting  of  a  sight  for  a  given  range ;  the 
adjustment  of  the  sight  to  make  the  shot  fall  at  a  given  range ;  and  the  sight  adjust- 
ments necessary  to  make  all  the  guns  of  a  battery  or  ship  shoot  together. 

PART  VI. 
Chapters  20  to  21  Inclusive. 

THE  accuracy  AND  PROBABILITY  OF  GUNFIRE  AND  THE  MEAN  ERRORS  OF  GUNS. 

The  errors  and  inaccuracies  of  guns.  The  probability  of  hitting  under  given 
conditions,  and  whether  or  not  it  would  be  wise  to  attempt  to  hit  under  these  con- 
ditions. The  number  of  shots  probably  necessary  to  give  a  desired  number  of  hits 
under  certain  given  conditions,  and  the  bearing  of  this  point  upon  the  wisdom  of 
attempting  an  attack  under  the  given  conditions,  having  in  mind  its  effect  upon  the 
total  amount  of  ammunition  available.  The  probabilities  governing  the  method  of 
spotting  salvos  by  maintaining  a  proper  number  of  "  shorts." 

9.  No  claim  is  made  to  originality  in  any  part  of  this  revision;  it  is  merely  a 
compilation  of  what  is  thought  to  be  the  best  and  most  modern  practice  from  the 
works  of  two  noted  investigators  of  the  subject,  namely.  Professor  Philip  R.  Alger, 
U.  S.  Navy,  and  Colonel  James  M.  Ingalls,  TJ.  S.  Army.  This  revised  work  is  based 
on  Professor  Alger's  text  book  on  Exterior  Ballistics  (edition  of  1906),  and  the 
additions  to  it  concerning  the  Ingalls  methods  are  from  the  Handbook  of  Problems 
in  Exterior  Ballistics,  Artillery  Circular  N,  Series  of  1893,  Adjutant  General's  Office 
(edition  of  1900),  prepared  by  Colonel  Ingalls.  Further  information  has  been  taken 
from  Bureau  of  Ordnance  Pamphlet  No.  500,  on  the  Methods  of  Computing  Range 
Tables.  The  two  chapters  on  the  calibration  of  guns  were  taken  from  a  pamphlet  on 
that  subject  written  by  Commander  L.  M.  Nulton,  U.  S.  Navy,  for  use  in  the  instruc- 
tion of  midshipmen,  and  Chapter  15  was  furnished  by  officers  on  duty  at  the  Naval 
Proving  Ground  at  Indian  Head. 

10.  To  summarize,  it  may  be  said  that  the  belief  is  held  that  the  only  reason  for 
teaching  the  science  of  exterior  ballistics  to  midshipmen  and  the  only  reason  for 
expecting  officers  to  possess  a  knowledge  of  its  principles  is  in  order  that  they  may 
intelligently  and  successfully  use  the  guns  committed  to  their  care.  So  much  as  is 
necessary  for  this  purpose  is  therefore  to  be  taught  the  undergraduate  and  no  more ; 
and,  as  the  information  necessary  for  the  scientific  use  of  a  gun  is  contained  in  the 
official  range  table  for  that  gun,  it  may  be  said  that  this  revision  of  the  previous  text 
book  on  the  subject  has  been  founded  upon  the  question : 

"  What  is  a  range  table,  how  is  the  information  contained  in  it  obtained,  and 
how  is  it  used  ?  " 
With  a  very  few  necessary  and  important  exceptions  such  as  the  computations  neces- 
sary in  determining  the  marking  of  sights,  the  text  of  the  book  follows  closely  the 
question  laid  down  above. 

11.  For  the  use  of  the  midshipmen  in  connection  with  this  revised  text  book,  a 
reprint  has  been  made  of  Table  II  for  the  desired  initial  velocities,  from  the  Ballistic 
Tables  computed  by  Major  J.  M.  Ingalls,  U.  S.  Army,  Artillery  Circular  M,  1900; 
a  reprint  of  the  table  from  Bureau  of  Ordnance  Pamphlet  No.  500,  for  use  in  con- 
nection with  Column  12  of  the  Range  Tables;  and  a  partial  reprint  of  the  Range 


PREFACE  9 

Tables  for  Naval  Guns  issued  by  the  Bureau  of  Ordnance;  these  being  in  addition  to 
the  tables  previously  available  for  midshipmen  in  the  older  text  book.  It  is  believed 
that  these  reprints  will  render  available  a  range  of  practical  problems  far  exceeding 
anything  that  has  heretofore  been  possible  for  the  instruction  of  midshipmen,  and 
that  by  their  use  they  will  be  graduated  and  commissioned  with  a  far  wider  knowledge 
of  the  meaning  and  use  of  these  tables,  and  of  the  underlying  principles  of  naval 
gunnery  than  has  ever  been  the  case  before. 

12.  The  work  of  revision  was  done  by  the  Head  of  Department,  assisted  by 
certain  officers  of  the  Department,  and  by  criticisms  and  suggestions  from  numerous 
other  officers,  on  duty  elsewhere  as  weJl  us  at  the  Naval  Academy..  Thanks  are  due 
to  all  those  who  helped  in  the  work,  and  especially  to  Lieutenant  (j.  g.)  C.  T.  Osburn, 
U.  S.  N.,  Lieutenant  (j.  g.)  W.  S.  Farber,  U.  S.  N".,  and  Lieutenant  (j.  g.)  N".  L, 
Nichols,  U.  S.  N.  Lieutenant  Osbum  carefully  scrutinized  the  text,  checked  all 
sample  problems  worked  in  the  text,  and  independently  worked  and  checked  the 
results  of  all  the  examples  given  at  the  ends  of  the  chapters  Lieutenants  Farber  and 
Nichols  checked  all  the  solutions  given  in  the  appendix,  and  Lieutenant  Nichols 
prepared  the  drawing  for  all  the  figures. 

L.  H.  CHANDLER, 
Captain,  U.  S.  Navy,  Head  of  Department. 


CONTENTS. 

PEELIMINAEY. 

CHAPTER  PAGE 

Introductory  Order   5 

Table  of  Symbols  Employed 13 

Table  of  Letters  of  Greek  Alphabet  Employed  as  Symbols 17 


PART  I. 
General  and  Approximate  Deductions. 

Introduction  to  Part  1 19 

1.  Definitions  and  Introductory  Explanations 21 

2.  The  Equation  to  the  Trajectory  in  a  Non-Resisting  Medium  and  the  Theory  of 

the  Rigidity  of  the  Trajectory  in  Vacuum 27 

3.  The  Resistance  of  the  Air,  the  Retardation  Resulting  Therefrom,   and   the 

Ballistic  Coefficient  35 

4.  The  Equation  to  the  Trajectory  in  Air  when  Mayevski's  Exponent  is  Equal  to  2.     47 

5.  Approximate  Determination  of  the  Values  of  the  Elements  of  the  Trajectory  in 

Air  when  Mayevski's  Exponent  is  Equal  to  2.     The  Danger  Space  and  the 
Computation  of  the  Data  Contained  in  Column  7  of  the  Range  Table 54 


PART  II. 
Practical  Methods. 

Introduction  to  Part  II 67 

6.  The  Time  and  Space  Integrals;  the  Computation  of  their  Values  for  Different 

Velocities,  and  their  Use  in  Approximate  Solutions 69 

7.  The  Differential  Equations  Giving  the  Relations  Between  the  Several  Elements 

of  the  General  Trajectory  in  Air.  Siacci's  Method.  The  Fundamental 
Ballistic  Formulae.  The  Computation  of  the  Data  Given  in  the  Ballistic 
Tables  and  the  Use  of  the  Ballistic  Tables 80 

8.  The  Derivation  and  Use  of  Special  Formulae  for  Finding  the  Angle  of  Departure, 

Angle  of  Pall,  Time  of  Flight,  and  Striking  Velocity  for  a  Given  Horizontal 
Range  and  Initial  Velocity;  that  is,  the  Data  Contained  in  Columns,  2,  3, 
4  and  5  of  the  Range  Tables.     Ingalls'  Methods 91 

9.  The  Derivation  and  Use  of  Special  Formulae  for  Finding  the  Coordinates  of  the 

Vertex  and  the  Time  of  Flight  to  and  the  Remaining  Velocity  at  the  Vertex, 
for  a  Given  Angle  of  Departure  and  Initial  Velocity,  which  Includes  the 
Data  Contained  in  Column  S  of  the  Range  Tables Ill 

10.  The  Derivation  and  Use  of  Special  Formula  for  Finding  the  Horizontal  Range, 

Time  of  Flight,  Angle  of  Fall,  and  Striking  Velocity  for  a  Given  Angle  of 
Departure  and  Initial  Velocity 120 

11.  The  Derivation  and  Use  of  Special  Formulae  for  Finding  the  Angle  of  Elevation 

Necessary  to  Hit  a  Point  Above  or  Below  the  Level  of  the  Gun  and  at  a 
Given  Horizontal  Distance  from  the  Gun,  and  the  Time  of  Flight  to,  and 
Remaining  Velocity  and  Striking  Angle  at  the  Target;  Given  the  Initial 
Velocity    125 

12.  The  Effect  Upon   the  Range  of  Variations   in   the   other  Ballistic   Elements, 

which  Includes  the  Data  Contained  in  Columns  10,  11,  12  and  19  of  the 
Range  Tables    131 


12  CONTENTS 

PART  III. 
r,-rrAT,rn^T,                 The  Vaeiation  of  the  Tkajectory  from  a  Plane  Curve,  „.^„ 

Introduction  to  Part  III 143 

13.  Drift  and  the  Theory  of  Sights,  Including  the  Computation  of  the  Data  Con- 

tained in  Column  6  of  the  Range  Tables 145 

14.  The  Effect  of  Wind  Upon  the  Motion  of  the  Projectile.     The  Effect  of  Motion 

of  the  Gun  Upon  the  Motion  of  the  Projectile.  The  Effect  of  Motion  of  the 
Target  Upon  the  Motion  of  the  Projectile  Relative  to  the  Target.  The  Effect 
Upon  the  Motion  of  the  Projectile  Relative  to  the  Target  of  all  Three 
Motions  Combined.  The  Computation  of  the  Data  Contained  in  Columns 
13,  14,  15,  16,  17  and  18  of  the  Range  Tables 153 

15.  Determination  of  Jump.    Experimental  Ranging  and  the  Reduction  of  Observed 

Ranges    166 

PART  IV. 
Range  Tables;  Their  Computation  and  Use. 

Introduction  to  Part  IV 1G9 

16.  The  Computation  of  the  Data  Contained  in  the  Range  Tables  in  General,  and 

the  Computation  of  the  Data  Contained  in  Column  9  of  the  Range  Tables. .   171 

17.  The  Practical  Use  of  the  Range  Tables 180 

PART  V. 
The  Calibration  of  Single  Guns  and  of  a  Ship's  Battery. 

Introduction  to  Part  V 213 

18.  The  Calibration  of  a  Single  Gun 215 

19.  The  Calibration  of  a  Ship's  Battery 226 

PART  VI. 
The  Accuracy  and  Probability  of  Gunfire  and  the  Mean  Errors  of  Guns 

Introduction  to  Part  VI 231 

20.  The  Errors  of  Guns  and  the  Mean  Point  of  Impact.    The  Equation  of  Probability 

as  Applied  to  Gunfire  when  the  Mean  Point  of  Impact  is  at  the  Center  of  the 
Target    233 

21.  The  Probability  of  Hitting  when  the  Mean  Point  of  Impact  is  Not  at  the  Center 

of  the  Target.  The  Mean  Errors  of  Guns.  The  Effect  Upon  the  Total 
Ammunition  Supply  of  Efforts  to  Secure  a  Given  Number  of  Hits  Upon  a 
Given  Target  Under  Given  Conditions.  Spotting  Salvos  by  Keeping  a 
Certain  Proportionate  Number  of  Shots  as  "  Shorts  " 246 

APPENDIX  A. 
Forms  to  be  Employed  in  the  Solution  of  the  Principal  Examples  Given  in  this 
Text  Book   261 

APPENDIX  B. 
Description  of  the  Farnsworth  Gun  Error  Computer 315 

APPENDIX  C. 
The  Use  of  Arbitrary  Deflection  Scales  for  Gun  Sights •■■■   326 

Atmospheric  Density  Tables 331 


TABLE  OF  SYMBOLS  EIMPLOYED. 

1.  The  symbols  employed  in  this  book  are  given  in  the  following  table.  So  far 
as  possible  they  are  those  employed  by  the  computers  of  the  Bureau  of  Ordnance  in 
their  work  of  preparing  range  tables,  etc. ;  but  a  considerable  number  of  additional 
symbols  have  been  found  necessary  in  a  text  book  treatment  of  the  subject. 

2.  In  quite  a  number  of  cases  it  has  been  found  necessary  or  advisable  to  use  the 
same  symbol  to  represent  two  or  more  different  quantities ;  but  such  quantities  are,  as 
a  rule,  widely  diiferent  from  each  other,  and  a  reasonable  amount  of  care  will  easily 
prevent  any  confusion  from  arising  from  this  cause. 

3.  The  symbols  employed  are  : 

PERTAINING  TO  THE  TRAJECTORY  AS  A  PLANE  CURVE. 


SYMBOL, 
R' 
X' 

R 
X 

</> 

w 

p 

j 

V 

V 

Vo 

Vh 

Vv 

U 

u 

(x,  y) 

(J'o.  2/o) 

y 

t 

T 

to 


QUANTITY    REPRESENTED    UY    IT. 

.Range  in  yards  on  an  inclined  plane. 

.Range  in  feet  on  an  inclined  plane. 

.Horizontal  range  in  yards. 

.Horizontal  range  in  feet. 

.Angle  of  departure. 

.Angle  of  fall. 

.Angle  of  elevation. 

.Angle  of  projection. 

.Angle  of  position. 

.Angle  of  jump. 

.Remaining  velocity  in  foot-seconds  at  any  point  of  the  trajectory. 

.Initial  velocity  in  foot-seconds. 

.Remaining  velocity  at  the  vertex  in  foot-seconds. 

.Remaining  velocity  at  point  of  fall  (or  striking  velocity)  in  foot-seconds. 

.Horizontal  velocity  at  any  point  of  the  trajectory  in  foot-seconds. 

.Vertical  velocity  at  any  point  of  the  trajectory  in  foot-seconds. 

.Pseudo  velocity  at  any  point  of  the  trajectory  in  foot-seconds. 

.Pseudo  velocity  at  the  muzzle  of  the  gun  in  foot-seconds;  [7  =  F. 

.Pseudo  velocity  at  the  vertex  in  foot-seconds. 

.Pseudo  velocity  at  the  point  of  fall  in  foot-seconds. 

.  Coordinates  of  any  point  of  the  trajectory  in  feet. 

.Coordinates  of  the  vertex  in  feet. 

.Ordinate  of  the  vertex  in  feet;  Y  =  y^- 

.Elapsed  time  of  flight  from  the  muzzle  to  any  point  of  the  trajectory  in 

seconds. 
.Time  of  flight  from  muzzle  to  point  of  fall  in  seconds. 
.Time  of  flight  from  muzzle  to  vertex  in  seconds. 
.Angle  of  inclination  of  the  tangent  to  the  trajectory  at  any  point  to  the 

horizontal. 


PERTAINING  TO  THE  PROJECTILE. 

w.  . .  .Weight  of  the  projectile  in  pounds. 
Aio.  . .  .Variation  from  standard  in  weight  of  projectile  in  pounds. 
d. . .  .Diameter  of  the  projectile  in  inches, 
c. . .  .Coefficient  of  form  of  the  projectile. 

C. ..  .Ballistic  coefficient.     When  written  with  numerical  subscripts,  as  1,  2, 
3,  etc.,  the  several  symbols  represent  successive  approximate  values 
of  C  as  appearing  in  the  computations.     The  same  system  of  sub- 
scripts is  used  for  a  similar  purpose  with  a  number  of  other  symbols. 
)3.  . .  .The  Integration  factor  of  the  ballistic  coefficient;  normally  /3  =  1. 
E . . .  .Constant  part  of  the  ballistic  coefficient  for  any  given  projectile,  given  by 


the  formula  K  = 


w 


TABLE  OF  SYMBOLS  EMPLOYED 

PERTAINING  TO  ATMOSPHERIC  CONDITIONS. 
Sj. . . .  Standard  density  of  the  atmosphere,  in  work  taken  as  unity. 
5 . . . .  Density  of  the  atmosphere  at  the  time  of  firing,  and  subsequently  repre- 
senting the  ratio  -^  =  ,-  . 

5i  1 

/.  . .  .Altitude  factor  of  the  ballistic  coefficient. 

PERTAINING  TO  WIND  AND  SPEEDS. 
W .  . .  .Real  wind;  force  in  feet  per  second. 
(S. . .  .Angle  between  wind  ahd  line  of  fire. 
Wx- . .  .Component  of  W  in  line  of  fire  in  feet  per  second. 
Wiox. . .  .Wind  component  in  feet  per  second  of  12  knots  in  line  of  fire. 

Wz Component  of  W  perpendicular  to  line  of  fire  in  feet  per  second. 

Wi2«. . .  .Wind  component  in  feet  per  second  of  12  knots  perpendicular  to  the  line 
of  fire. 
X. . .  .Range  in  feet  without  considering  wind. 
X' . . .  .Range  in  feet  considering  wind. 

T . . .  .Initial  velocity  in  foot-seconds  without  considering  wind, 
y .  . .  .  Initial  velocity  in  foot-seconds  considering  wind. 
(^.  . .  .Angle  of  departure  without  considering  wind. 
0'.  . .  .Angle  of  departure  considering  wind. 
T. . .  .Time  of  fiight  in  seconds  without  considering  wind. 
T' .  .  .  .Time  of  flight  in  seconds  considering  wind. 
AXir-  . .  .Variation  in  range  in  feet  due  to  Wj- 
AXj,,r-  ■•  .Variation  in  range  in  feet  due  to  a  wind  component  of  12  knots  in  the 
line  of  fire. 
ARffr. . .  .Variation  in  range  in  yards  due  to  Wj. 
ARiiW-  ■  ■  .Variation  in  range  in  yards  due  to  a  wind  component  of  12  knots  in  the 
line  of  fire. 
7.... Angle  between  the  trajectories  relative  to  the  air  and  relative  to  the 
ground. 
Dw  ■  ■  .Deflection  in  yards  due  to  W^. 
Dy.w  ■  ■  .Deflection  in  yards  due  to  a  wind  component  of  12  knots  perpendicular  to 
the  line  of  fire. 
G. . .  .Motion  of  gun  in  feet  per  second. 
Gx-  ■  ■  .Component  of  the  motion  of  the  gun  in  the  line  of  fire  in  feet  per  second. 
G,2a;.  . .  .Motion  of  gun  in  line  of  fire  in  feet  per  second  for  a  component  of  motion 
of  gun  in  that  line  of  12  knots. 
Gz-  ■  ■  .Component  of  the  motion  of  the  gun  perpendicular  to  the  line  of  fire  in 
feet  per  second. 
Gi22.  . .  .Motion  of  gun  perpendicular  to  line  of  fire  in  feet  per  second  for  a  com- 
ponent of  motion  of  gun  of  12  knots  in  the  same  direction. 
AXq.  ..  .Variation  in  range  in  feet  resulting  from  G^. 
AXyQ.  ..  .Variation  in  range  in  feet  due  to  a  motion  of  the  gun  of  12  knots  in  the 
line  of  fire. 
Ai?o.  . .  .Variation  in  range  in  yards  resulting  from  Gx- 
Aliv.G-  ■■  .Variation  in  range  in  yards  due  to  a  motion  of  the  gun  of  12  knots  in  the 
line  of  fire. 
Do. ..  .Deflection  in  yards  due  to  G-. 
Z)i„f;.  ..  .Deflection  in  yards  due  to  the  motion  of  the  gun  of  12  knots  perpen- 
dicular to  the  line  of  flre. 
T. . .  .Motion  of  target  in  feet  per  second. 

Tj. ..  .Component  of  the  motion  of  the  target  in  the  line  of  flre  in  feet  per 
second. 
Tjoj. ....  Motion  of  target  in  line  of  fire  in  feet  per  second  for  a  component  of 
motion  of  target  in  that  line  of  12  knots. 
T-. ..  .Component  of  the  motion  of  the  target  perpendicular  to  the  line  of  fire 
in  feet  per  second. 
Ti2« ....  Motion  of  target  perpendicular  to  line  of  fire  in  feet  per  second  for  a 
component  of  motion  of  target  of  12  knots  in  the  same  direction. 


TABLE  OF  SYMBOLS  EMPLOYED  15 

AJr-  •  •  .Variation  in  range  in  feet  resulting  from  Tj-. 
AXi2T.  ■  ■  .Variation  in  range  in  feet  due  to  a  motion  of  the  target  of  12  knots  in  tlie 
line  of  fire. 
ARt-  ■  ■  .Variation  in  range  in  yards  resulting  from  Tx- 
ARi2T-  ■  ■  .Variation  in  range  in  yards  due  to  a  motion  of  the  target  of  12  knots  in 
the  line  of  fire. 
Dt-  ■■  .Deflection  in  yards  due  to  Tg. 
Di2T-  •■  .Deflection  in  yards  due  to  a  motion  of  the  target  of  12  knots  perpen- 
dicular to  the  line  of  fire, 
a.  . .  .Angle  of  real  wind  with  the  course  of  the  ship, 
a' . . . .  Angle  of  apparent  wind  with  the  course  of  the  ship. 
W .  .  .  .Velocity  of  the  real  wind  in  knots  per  hour. 
W".  . .  .Velocity  of  the  apparent  wind  in  knots  per  hour. 

PERTAINING  TO  THE  THEORY  OF  PROBABILITY. 
X.  . .  .Axis  of;  axis  of  coordinates  lying  along  range,  for  points  over  or  short 

of  the  target. 
Y.  .  .  .Axis  of;  axis  of  coordinates  in  vertical  plane  through  target  for  points 

above  or  below  the  center  of  the  target. 
Z....Axis  of;  axis  of  coordinates  in  vertical  plane  through  target  for  points 
to  right  or  left  of  the  center  of  target. 
(^1,  Vi),  etc. . .  .Coordinates  of  points  of  impact  in  vertical  plane  through  target. 
^z.  . .  .  Summation  of  2;,,  z^,  etc. 
2y.  . .  .Summation  of  1/1,  y-,,  etc. 
«...  .Number  of  shot. 

yz. . .  .Mean  deviation  along  axis  of  Z,  that  is,  above  or  below. 
yy.  . .  .Mean  deviation  along  axis  of  Y,  that  is,  to  right  or  left. 
yx-  ■  ■  .Mean  deviation  along  axis  of  X,  that  is,  in  range. 

P. . .  .Probability  that  the  deviation  of  a  single  shot  will  be  numerically  less 
than  the  given  quantity  a. 

—  . ..  .Argument  for  probability  table. 
7 

PERTAINING  TO  VARIATIONS  IN  THE  BALLISTIC  ELEMENTS. 

AX. . .  .Variation  in  the  range  in  feet. 
AR. . .  .Variation  in  the  range  in  yards. 
A(sin  2^) . . .  .Variation  in  the  sine  of  twice  the  angle  of  departure. 

ArA-  ■  •  .Quantity  appearing  in  Table  II  of  the  Ballistic  Tables  in  the  Ar  column 
pertaining  to  "  A."  With  figures  before  the  subscript  Y  it  shows  the 
amount  of  variation  in  V  for  which  used.  (Be  careful  not  to  con- 
fuse this  symbol  with  AV  or  dV.) 
SY . . .  .Variation  in  the  initial  velocity.  (Be  careful  not  to  confuse  this  symbol 
with  A,-,i  or  AV.) 

AV.  . .  .Difference  between  V  for  two  successive  tables  in  Table  II.  (Be  careful 
not  to  confuse  this  symbol  with  Ay^  or  dV.) 

AY j<,. ..  .Variation  in  the  initial  velocity  in  foot-seconds  due  to  variation  in  the 
weight  of  the  projectile  in  pounds.  Figures  before  the  to  show  the 
amount  of  variation  in  that  quantity  in  pounds. 

AX;r.  ..  .Variation  in  range  in  feet  due  to  a  variation  in  V  in  foot-seconds. 
Figures  before  the  Y  show  the  amount  of  variation  in  that  quantity 
in  foot-seconds. 

Ai^r- ■• -Variation  in  range  in  yards  due  to  a  variation  in  Y  in  foot-seconds. 
Figures  before  the  Y  show  the  amount  of  variation  in  that  quantity 
in  foot-seconds. 
AC  . .  .Variation  in  the  ballistic  coefl^cient  in  percentage. 

AXq.  . .  .Variation  in  range  in  feet  due  to  a  variation  in  C  in  percentage.  Figures 
before  the  C  show  the  amount  of  variation  in  that  quantity  in  per- 
centage. 

Ai?c'- ■• -Variation  in  range  in  yards  due  to  a  variation  in  C  in  percentage. 
Figures  before  the  C  show  the  amount  of  variation  in  that  quantity 
in  percentage. 


16  TABLE  OF  SYMBOLS  EMPLOYED 

A5. . .  .Variation  in  S  in  percentage. 

Ai'g Variation  in  range  in  feet  due  to  a  variation  in  5  in  percentage.    Figures 

before  the  5  sliow  the  amount  of  variation  in  that  quantity  in  per- 
centage. 
ARg ....  Variation   in   range  in  yards  due  to  a  variation   in   5   in   percentage. 
Figures  before  the  5  show  the  amount  of  variation  in  that  quantity 
in  percentage. 
Aw.  . .  .Variation  in  w  in  pounds. 

AXy, Variation  in  range  in  feet  due  to  a  variation  in  w  in  pounds.     Figures 

before  the  w  show  the  amount  of  variation  in  that  quantity  in 
pounds. 

AX'w That  part  of  AXw  in  feet  which  is  due  to  the  reduction  in  initial  velocity 

resulting  from  Aio. 

AX"w That  part  of  AXy^  in  feet  which  is  due  to  Am;  directly. 

Ai?,„ Variation  in  range  in  yards  due  to  a  variation  in  w  in  pounds.    Figures 

before  the  w  show  the  amount  of  variation  in  that  quantity  in 
pounds. 

AR\, That  part  of  AR^  in  yards  which  is  due  to  the  reduction  in  initial 

velocity  resulting  from  A%o. 

AR"u, That  part  of  A72,„  in  yards  which  is  due  to  Aw  directly. 

H Change  in  height  of  point  of  impact  on  a  vertical  screen  through  the 

target,  in  feet,  due  to  a  change  of  AR  in  R  in  yards.  Figures  as  sub- 
scripts to  the  H  show  the  change  in  R  necessary  to  give  that  par- 
ticular value  of  H. 

MATHEMATICAL  AND  MISCELLANEOUS. 

g Acceleration  due  to  gravity  in  foot-seconds  per  second;  g  =  32.2. 

dx. . .  .Differential  increment  in  x. 

dy...  .Differential  increment  in  y. 

ds. . .  .Differential  increment  along  the  curve,  that  is,  in  s. 

dv . . .  .Differential  increment  in  v. 

dt. . .  .Differential  increment  in  t. 

du.  . .  .Differential  increment  in  u. 

a. . .  .Mayevski's  exponent. 

A Mayevski's  coefficient. 

R. . .  .Total  air  resistance  in  pounds. 

Rf...  .Total  air  resistance  in  pounds  under  firing  conditions. 
iSs--- -Total  air  resistance  in  pounds  under  standard  conditions, 
p. . .  .Radius  of  curvature  of  the  trajectory  at  any  point  in  feet. 

fc. . .  .The  value  of -tt  ,  in  which  A  is  Mayevski's  constant  and  C  is  the  ballistic 

coefficient. 
e. . .  .The  base  of  the  Naperian  system  of  logarithms;  e  =  2.7183. 
n. . .  .The  ratio  between  the  range  in  vacuum  and  the  range  in  air  for  the  same 
angle  of  departure. 
Ty. . .  .Value  of  the  time  integral  in  seconds  for  remaining  velocity  v. 
Tv  ■■  .Value  of  the  time  integral  in  seconds  for  initial  velocity  Y. 
Sv  ■  ■  .Value  of  the  space  integral  in  feet  for  remaining  velocity  v. 
*SV. . .  .Value  of  the  space  integral  in  feet  for  initial  velocity  V. 
Tji. . .  .Value  of  the  time  function  in  seconds  for  pseudo  velocity  u. 
Tv  ■  ■  .Value  of  time  function  in  seconds  for  initial  velocity  V. 
8u-  •  •  .Value  of  space  function  in  feet  for  pseudo  velocity  u. 
Sy  ■■  .Value  of  space  function  in  feet  for  initial  velocity  V. 
Au-  •  ■  .Value  of  altitude  function  for  pseudo  velocity  u. 
Ar-  ■■  .Value  of  altitude  function  for  initial  velocity  V. 
lu. . .  .Value  of  inclination  function  for  pseudo  velocity  u. 
Iy_ ..  .Value  of  inclination  function  for  initial  velocity  V. 
Su   ...  .Value  of  space  function  in  feet  for  pseudo  velocity  u^. 
Su  .  . .  .Value  of  space  function  in  feet  for  pseudo  velocity  m„. 
Tu   ...  .Value  of  time  function  in  seconds  for  pseudo  velocity  u  . 


TABLE  OF  SYMBOLS  EMPLOYED 


17 


0 

/u„. 
AS. 
AT. 
AA. 

AI. 
z. 


a.  h,  a',  t' . . 
A,  B.  A',  T'..  . 

A  '  and  B  =  -r 
A 

<t>x.  ■ 


n 

n 
D' 
D 
I 
D 
d 


h 
E 
K 
K' 

n 

S 
a 

7 
(a.  a') 
(b,  6') 
(c,  c') 
(.d,d') 


.  .Value  of  time  function  in  seconds  for  pseudo  velocity  Uo. 

.  .Value  of  altitude  function  for  pseudo  velocity  u  . 

.  .Value  of  altitude  function  for  pseudo  velocity  u„. 

.  .Value  of  inclination  function  for  pseudo  velocity  u  . 

.  .Value  of  inclination  function  for  pseudo  velocity  Uo. 

. .  Difference  between  two  values  of  the  space  function. 

. .  Difference  between  two  values  of  the  time  function. 

. .  Difference  between  two  values  of  the  altitude  function. 

. .  Difference  between  two  values  of  the  inclination  function. 

. .  General  expression  for  value  of  argument  in  Column  1  of  Table  II  of  the 

Ballistic  Tables,  for  any  point  of  the  trajectory;  2;  =  -^  . 
.  .Special  expression  for  value  of  argument  in  Column  1  of  Table  II  of  the 

Ballistic  Tables,  for  the  whole  trajectory;  Z  =  -^. 
.  General  values  of  Ingalls'  secondary  functions. 


i  Special  value  of  Ingalls'  secondary  functions  for  whole  trajectory. 

.  .Angle  of  departure  for  a  horizontal  distance  x. 

.  .A  ratio  used  in  computing  the  drift;  in  which  k  is  the  radius  of  gyration 
of  the  projectile  about  its  longitudinal  axis,  and  R  is  the  radius  of 
the  projectile. 

.  .A  special  ratio  used  in  computing  the  drift. 

. .  The  twist  of  the  rifling,  used  in  computing  the  drift. 

.  .Ingalls'  secondary  function  for  drift. 

.  .Drift  in  yards. 

.  .Sight  radius  in  inches. 

.  .Deflection  in  yards  (used  with  R  in  yards  in  deflection  computations). 

. .  Distance  in  inches  which  the  sliding  leaf  is  set  over  to  compensate  for 

the,  deflection  D  in  yards. 
. .  Permanent  sight-bar  angle. 
.  .  Sight  bar  height  in  inches. 
.  .Penetration  of  armor  in  inches. 
.  .Constant  used  in  computing  penetration  of  armor. 
.  .Constant  used  in  computing  penetration  of  armor. 
. .  Height  of  target  in  feet. 
.  .Danger  space  in  general. 

•  |-.\ngles  for  plotting  fall  of  shot  in  calibration  practice. 


Coordinates  of  point  of  fall  of  shot  for  plotting  in  calibration  practice. 


LETTERS  OF  THE  GREEK  ALPHABET  USED  AS  SYMBOLS. 

Letter.  Pronunciation. 

.Alpha. 
.Beta. 
7.  . .  .Gamma. 
A  or  S. . .  .Delta. 

.  .Epsilon. 


Letter. 

Pronunciation. 

Lett€ 

r. 

Pronunciation. 

e. . 

..Theta. 

S  or 

a. 

. . .  Sigma. 

X.. 

.  .Lambda. 

<*. 

...Phi. 

fj..  . 

..Mu. 

^- 

...Psi. 

p.. 

..Pi. 
..Rho. 

u , 

. .  .Omega. 

PAET  I. 
GENERAL  AND  APPROXIMATE  DEDUCTIONS. 

INTRODUCTION  TO  PART  I. 

Part  I  of  this  text  book  includes  the  preliminary  definitions  and  discussions, 
from  which  we  pass  to  the  derivation  of  the  equation  to  the  trajectory  in  a  non- 
resisting  medium,  and  thence  to  the  principle  of  the  rigidity  of  the  trajectory  in 
vacuum. 

As  the  next  step  comes  the  discussion  of  the  resistance  of  the  atmosphere  to  the 
passage  of  a  projectile  through  it,  and  of  the  retardation  in  the  motion  of  the  pro- 
jectile resulting  from  this  resistance;  and  following  this,  a  consideration  of  the 
ballistic  coefficient  in  all  its  forms ;  and  following  this  again,  a  statement  of  Mayevski's 
general  and  special  formula  for  retardation,  that  is,  of  the  general  formula,  and  of 
the  numerical  results  of  experiments  to  determine  the  resistance  and  retardation. 

We  are  then  prepared  to  take  up  the  derivation  of  the  equation  to  the  trajectory 
in  air,  having  determined  the  forces  that  act  upon  a  projectile  in  flight.  We  will  find 
that  it  is  only  possible  to  determine  such  an  expression  for  certain  given  velocities, 
but  having  thus  determined  it,  we  will  be  able  to  get  a  comparison  between  the 
trajectories  in  vacuum  and  in  air  under  similar  conditions  (for  the  special  conditions 
under  which  we  were  able  to  derive  the  equation  to  the  trajectory  in  air).  We  can 
also  derive  certain  formulae  for  determining  approximate  values  of  the  elements  of 
the  trajectory  in  air,  by  the  use  of  the  equation  above  described. 

This  completes  the  preliminary,  educational  and  approximate  consideration  of 
the  trajectory  as  a  plane  curve,  and  enables  us  to  pass  on,  in  Part  II,  to  the  more 
practical  methods  actually  employed  by  artillerists  in  making  such  computations  as 
are  necessary  for  the  successful  use  of  guns.  It  must  be  noted,  however,  that  some  of 
the  approximate  methods  given  in  Part  I  give  results  that  are  sufficiently  accurate 
to  render  them  available  for  general  use,  and  that  these  results  are  actually  so  used 
in  practice. 


CHAPTEE  1. 


DEFINITIONS  AND  INTRODUCTORY  EXPLANATIONS. 

Symbols  Introduced. 

B' .  . .  .  Eange  in  yards  on  an  inclined  plane. 
X' .  . . .  Range  in  feet  on  an  inclined  plane. 
11.  . . .  Horizontal  range  in  yards. 
X .  . .  .  Horizontal  range  in  feet.  • 

(^.  . .  .Angle  of  departure. 

tu .  . .  .  Angle  of  fall. 

p.  . .  .  Angle  of  position. 

;'....  Angle  of  jump. 

i// .  . .  .  Angle  of  elevation. 
i//' .  . .  .  Angle  of  projection. 
V .  . . .  Initial  velocity  in  foot-seconds. 

V. . .  .Eemaining  velocity  at  any  point  in  the  trajectory  in  foot-seconds, 
t'o .  . .  .  Eemaining  velocity  at  the  vertex  in  foot-seconds. 
Vcj.  . .  .Eemaining  velocity  at  the  point  of  fall,  or  striking  velocity,  in  foot- 
seconds. 
Vh ....  Horizontal  velocity  at  any  point  of  the  trajectory  in  foot-seconds. 
Vf.  . .  .Vertical  velocity  at  any  point  of  the  trajectory  in  foot-seconds. 

u.  . .  .Pseudo  velocity  at  any  point  of  the  trajectory  in  foot-seconds. 
U .  . . .  Pseudo  velocity  at  the  muzzle  of  the  gun  in  foot-seconds;  U=  V, 
Vq.  .  .  .Pseudo  velocity  at  the  vertex  in  foot-seconds. 
Uu.  . . .  Pseudo  velocity  at  the  point  of  fall  in  foot-seconds. 

1.  Ballistics  is  the  science  of  the  motion  of  projectiles,  and  is  divided  into  two  Definitions, 
branches ;  namely,  interior  ballistics  and  exterior  ballistics. 

2.  Interior  ballistics  is  that  branch  of  the  science  which  treats  of  the  motion  of 
the  projectile  while  in  the  gun  and  of  the  phenomena  which  cause  and  attend  this 
motion. 

3.  Exterior  ballistics  is  that  branch  of  the  science  which  treats  of  the  motion  of 
the  projectile  after  it  leaves  the  gun.  The  investigations  to  be  conducted  under 
exterior  ballistics  therefore  begin  at  the  instant  when  the  projectile  leaves  the  muzzle 
of  the  ffuu. 


22 


EXTERIOE  BALLISTICS 


4.  There  are  certain  definitions  connected  with  the  travel  of  the  projectile  after 
it  leaves  the  gun,  and  certain  symbols  which  are  used  to  represent  the  quantities 
covered  by  these  definitions.    These  definitions  and  symbols  will  now  be  given. 


Figure  1. 


Elements  of  Tbajectort. 


BOII:^<f>  =  Ang]e  of  Departure. 

fl^=:  Point  of  Fall,  Horizontal  Range. 
DnO  =  w  =  Ang\e  of  Fall. 

Oi/ =3  X=  Horizontal  Range. 

Oi/ ^Line  of  Position. 
JI/Off:=p  =  Angle  of  Position. 
D'ME=:w'  —  Ansle  of  Fall. 


OJ/  =  X'  =  Range. 
50if  :=!/''=:  Angle  of  Projection. 
BOA^j:^ Angle  of  jump. 
AOM  =  \p  =  Angle  of  Elevation. 

<p  =  i^  +  j  +  p  =  i''  +  p- 

When  1)^0.     <p  =z ■^' =  ij/  -\-  j , 
When  p^j^O,  (p=i^=z\p'. 


5.  The  trajectory  is  the  curve  traced  by  the  projectile  in  its  flight  from  the 
muzzle  of  the  gun  to  the  first  point  of  impact,  which  point  of  impact  is  called  the 
point  of  fall;  in  other  words,  it  is  the  path  of  the  projectile  between  those  two  points 
considered  as  a  curve. 

6.  The  elements  of  the  trajectory,  broadly  considered,  are  certain  quantities 
which  are  now  to  be  defined,  such  as  the  initial  velocity,  range,  time  of  flight,  etc., 
which  enter  into  the  mathematical  consideration  of  the  trajectory. 

7.  The  range  is  the  distance  in  a  straight  line  from  the  gun  to  the  point  of  fall. 
It  will  be  denoted  by  R'  when  given  in  yards  and  by  X'  when  given  in  feet. 

8.  When  the  point  of  fall  is  in  the  same  horizontal  plane  as  the  gun,  the  range  is 
called  the  horizontal  range.  It  will  be  denoted  by  R  when  given  in  yards  and  by  X 
when  given  in  feet.  Unless  otherwise  stated,  in  discussions  and  problems,  the  term 
range  will  always  mean  horizontal  range. 

9.  The  line  of  departure  is  the  line  in  which  the  projectile  is  moving  when  it 
leaves  the  gun.  It  is  tangent  to  the  trajectory  at  its  origin,  and,  with  modern  guns, 
it  practically  coincides  with  the  axis  of  the  bore  of  the  gun. 

10.  The  angle  of  departure  is  the  angle  between  the  tangent  to  the  trajectory  at 
the  origin,  that  is  the  line  of  departure,  and  the  horizontal  plane.  It  will  be  denoted 
bysb. 


GENERAL  AND  APPEOXIMATE  DEDUCTIONS        23 

11.  The  angle  of  fall  is  the  angle  between  the  tangent  to  the  trajectory  at  the 
point  of  fall  and  the  horizontal  plane.    It  will  be  denoted  by  w. 

12.  The  line  of  position  is  the  straight  line  from  the  gun  to  the  target.  As  it 
coincides  with  the  line  of  vision  through  the  sight  when  the  gun  is  properly  aimed, 
it  is  sometimes  called  the  line  of  sight. 

13.  The  angle  of  position  is  the  angle  between  the  line  of  position  and  the  hori- 
zontal plane.  It  will  be  denoted  by  p;  and  is  positive  when  the  target  is  higher  than 
the  gun,  and  negative  when  the  target  is  lower  than  the  gun. 

14.  The  angle  of  elevation  is  the  angle  between  the  axis  of  the  bore  of  the  gun 
and  the  line  of  position  at  the  instant  before  firing.    It  will  be  denoted  by  i/^. 

15.  The  angle  of  projection  is  the  angle  between  the  axis  of  the  bore  of  the  gun 
and  the  line  of  position  at  the  instant  the  projectile  leaves  the  muzzle.  It  will  be 
denoted  by  i/''.        ~~~"^-~---~___ 

16.  The  angle  of  jump  or  jump  is  the  difference  between  the  angle  of  projection 
{yj/')  and  the  angle  of  elevation  (i/^)  ;  in  other  words,  it  is  the  vertical  angle  which  the 
axis  of  the  gun  describes  under  the  shock  of  firing  during  the  interval  from  the 
ignition  of  the  powder  charge  to  the  exit  of  the  projectile  from  the  gun.  It  will  be 
denoted  by  ;";  positive  when  the  muzzle  of  the  gun  jumps  upward,  and  negative  when 
it  jumps  downward.    Jump  is  generally  positive,  but  not  invariably  so. 

17.  The  muzzle  velocity  or  initial  velocity  is  the  velocity  with  which  the  pro-  initial 
jectile  is  supposed  to  leave  the  muzzle  of  the  gun.    It  is  measured  in  feet  per  second,    ^  °°*  ^' 
foot-seconds  (denoted  by  f.  s.),  and  is  often  indicated  by  the  abbreviation  I.  V.     It 

will  be  denoted  by  V.  As  it  is  impracticable  in  practice,  for  reasons  that  will  be 
readily  apparent,  to  actually  measure  the  velocity  of  the  projectile  at  the  instant  it 
leaves  the  muzzle,  it  is  the  custom  to  determine  it  by  actual  measurements  at  a  known 
distance  from  the  gun,  that  distance  being  as  small  as  possible  and  still  keep  the 
necessary  measuring  apparatus  from  being  destroyed  by  the  blast  from  the  gun. 
Having  determined  the  velocity  at  this  known  short  distance  from  the  gun,  it  is  the 
custom  to  then  compute,  by  formulae  to  be  derived  later,  what  the  velocity  would 
have  been  at  the  muzzle  had  the  conditions  at  the  muzzle  and  from  the  muzzle  to  the 
point  of  measurement  been  the  same  as  at  that  latter  point.  Actually,  however,  the 
projectile,  immediately  after  leaving  the  muzzle,  is  surrounded  by  gases  moving  even 
more  rapidly  than  the  projectile  itself,  and  experiments  have  shown  that  the  velocity 
increases  instead  of  decreases  during  a  travel,  which,  for  large  guns,  may  be  as  much 
as  fifty  yards  after  leaving  the  gun.  Thus  the  initial  or  muzzle  velocity  used  in 
exterior  ballistics  is  really  a  fictitious  quantity  not  actually  existent  when  the  pro- 
jectile leaves  the  muzzle,  but  which  may  be  more  accurately  defined  as  the  velocity 
with  which  it  would  be  necessary  to  project  the  projectile  from  the  muzzle  into  still 
air  in  order  to  have  it  describe  the  actual  trajectory. 

18.  The  remaining  velocity  at  any  point  in  the  trajectory-  is  the  actual  velocity 
in  foot-seconds  at  that  point  in  a  line  which  is  tangent  to  the  trajectory  at  that  point. 
It  will  be  denoted  by  v.    At  the  vertex  it  is  denoted  by  v^. 

19.  The  striking  velocity  is  the  remaining  velocity  at  the  point  of  fall.  It  will 
be  denoted  by  v^^. 

20.  The  horizontal  velocity  at  any  point  in  the  trajectory  is  the  horizontal  com- 
ponent of  the  remaining  velocity  at  that  point.    It  will  be  denoted  by  Vjt. 

21.  The  vertical  velocity  at  any  point  of  the  trajectory  is  the  vertical  component 
of  the  remaining  velocity  at  that  point.    It  will  be  denoted  by  Vv. 

22.  The  pseudo  velocity  at  any  point  of  the  trajectory  is  the  component  of  the 
remaining  velocity  at  that  point  in  a  line  parallel  to  the  line  of  departure.  It  is  a 
most  important  quantity  in  ballistic  computations,  and  will  be  denoted  by  u.    At  the 


24  EXTEEIOR  BALLISTICS 

muzzle  it  becomes  U,  and  is  equal  to  V ;  at  the  vertex  it  becomes  Wq  ;  and  at  the  point 
of  fall  it  becomes  u^^. 

23.  The  two  following  assumptions,  which  are  sufficiently  correct  for  all  present 
practical  purposes,  are  made  throughout : 

1.  The  force  of  gravity  throughout  the  trajectory  acts  in  parallel  lines  per- 
pendicular to  the  horizontal  plane  at  the  gun  ;  the  value  of  g  being  32.2  f .  s.  s. 

2.  The  dimensions  of  the  gun  are  negligible  in  comparison  with  the  trajectory. 
For  convenience  we  may  therefore  suppose  the  trajectory  to  begin  at  the  axis  about 
which  the  gun  is  being  elevated  or  depressed,  and  we  will  take  the  horizontal  plane 
through  that  axis  to  be  the  horizontal  plane  through  the  gun, 

24.  Figure  1  represents  a  trajectory,  0  indicating  the  position  of  the  gun  or 
origin,  and  OH  the  horizontal  plane.  Then  BOn=(f>  is  the  angle  of  departure ;  H  the 
point  of  fall  on  the  horizontal  plane;  DHO  =  w  is  the  angle  of  fall;  and  OH  =  X  (or 
E)  is  the  horizontal  range.  If  the  target  be  at  M,  then  OM  is  the  line  of  position; 
MOH  =  p  is  the  angle  of  position;  D'ME  =  o/  is  the  angle  of  fall;  and  OM  =  X'  (or 
R')  is  the  range.  BOM  =  xl/'  is  the  angle  of  projection,  which  coincides  with  the  angle 
of  departure,  </>,  when  the  angle  of  position,  p,  is  zero.  OA  represents  the  position  of 
the  axis  of  the  bore  at  the  instant  before  firing,  and  OB  its  position  at  the  instant  the 
projectile  leaves  the  muzzle,  the  angle  between  them,  BOA=i,  being  the  a;\?le  of 
jump.  The  angle  AOM  =  xp  is  the  angle  of  elevation.  It  will  be  seen  that  <f)  =  i]/+j 
-^p  =  if/'-\-pj  and  that  when  p  =  0  this  becomes  <f,  =  \p'  =  \p-\-i. 

25.  In  studying  these  definitions  it  should  be  noted  that  in  them,  as  given  for  the 
various  angles,  the  angle  of  departure  (^)  and  the  angle  of  fall  (w)  are  the  only  ones 
that  are  measured  from  the  horizontal  plane,  except  that  of  course  the  position  angle 
{p)  is  by  its  very  definition  the  angle  between  the  line  of  position  and  that  plane. 
The  angle  of  elevation  {^)  and  the  angle  of  projection  (i/^')  are  measured  from  the 
line  of  position.  Of  course  when  we  are  working  with  horizontal  ranges,  which  is 
generally  the  case,  and  when  there  is  no  jump,  which  is  also  generally  the  case  (when 
p  =  Q  and  ;  =  0),  then  the  angle  of  elevation,  angle  of  projection,  and  angle  of 
departure  all  become  the  same,  that  is,  (f}  =  ij/  =  ij/'. 

26.  Being  now  familiar  with  certain  definitions  relating  to  the  trajectory,  we 
may  undertake  its  consideration  in  a  general  way.  If  it  were  possible  to  fire  the  gun 
and  have  the  whole  travel  of  the  projectile  take  place  in  a  non-resisting  medium,  as  in 
vacuum  for  instance,  it  is  apparent  that,  after  it  has  acquired  its  initial  velocity,  the 
only  force  acting  upon  the  projectile  during  its  fiight  is  the  force  of  gravity.  The 
derivation  of  the  equation  to  the  trajectory  in  vacuum  and  the  investigation  of  its 
elements  therefore  becomes  a  very  simple  matter,  as  will  be  seen  in  the  next  chapter. 

27.  It  is  also  apparent  that,  as  the  fiight  of  the  projectile  necessarily  takes  place 
in  a  resisting  medium,  that  is  the  atmosphere,  there  must  really  be  in  actual  practice, 
in  addition  to  the  force  of  gravity,  a  force  acting  upon  it  during  flight  due  to  the 
atmospheric  resistance.  Such  being  the  case,  it  is  evident  that  the  investigation  of 
the  trajectory  in  vacuum,  while  most  necessary  from  an  educational  standpoint,  must 
necessarily  be  of  comparatively  little  real  value  in  the  solution  of  practical  problems 
in  gunnery. 

28.  It  is  proposed,  in  this  text  book,  to  first  discuss  the  trajectory  in  vacuum, 
in  order  to  derive  from  it  such  general  knowledge  as  is  of  value  in  later  work.  Then 
an  equation  to  the  trajectory  in  air  will  be  derived  for  certain  special  conditions,  in 
order  that  it  may  be  compared  with  the  equation  to  the  trajectory  in  vacuum  for  the 
same  angle  of  departure  and  initial  velocity.  It  will  be  shown,  however,  that  neither 
of  the  above  equations  is  very  serviceable  for  the  solution  of  practical  problems  in 
service  gunnery  with  modern  velocities,  and  other  mathematical  formulas  will  be 


GENEEAL  AND  APPEOXIMATE  DEDUCTIONS 


25 


deduced  which,  while  not  general  equations  to  the  complete  trajectory  in  air  as  a 
whole,  will  nevertheless  express  the  relations  that  always  exist  between  certain  ele- 
ments of  that  trajectory.  These  formulae  w'ill  be  generally  true  for  all  velocities,  and 
will  be  in  such  form  that  practical  results  can  be  obtained  by  their  use. 

29.  These  expressions  having  been  deduced,  it  will  be  shown  how  they  are  used 
in  the  computation  of  the  data  contained  in  the  range  tables ;  and  thence,  conversely, 
how  the  data  contained  in  those  tables  can  be  used  in  service  aboard  ship. 

30.  The  assumption  that  the  trajectory  in  vacuum  is  a  plane  curve,  as  already 
stated,  is  based  upon  the  fact  that,  under  these  conditions  the  only  force  acting  upon 
the  projectile  in  flight  is  the  force  of  gravity,  and  as  this  force  acts  solely  in  the 
vertical  plane  through  the  projectile,  it  is  evident  that  there  is  no  force  present  to 
divert  the  projectile  from  the  original  plane  of  fire.  When  the  flight  takes  place  in 
air,  however,  in  addition  to  the  force  of  gravity,  we  have  the  atmospheric  resistance 
acting  to  retard  the  flight  of  the  projectile,  and  it  is  assumed  that  this  resistance  acts 
at  every  point  in  the  trajectory  along  the  tangent  to  the  curve  at  such  point.  This 
retarding  force  therefore  always  acts  in  the  original  vertical  plane  of  fire,  if  our 
assumption  be  correct,  and  therefore  the  trajectory  in  air  also  remains  a  plane  curve. 

31.  It  will  be  shown  later,  however,  that  this  assumption,  that  the  resistance  of 
the  air  always  acts  in  the  original  vertical  plane  of  fire,  is  in  error,  and  that  the 
trajectory  in  air  is  not  exactly  a  plane  curve ;  but  it  will  also  be  shown  that  the  errors 
in  the  computed  values  of  ranges,  angles,  times,  etc.,  based  on  formulae  derived  on  the 
assumption  that  the  trajectory  is  a  plane  curve,  are  so  small  that  such  results  may  be 
considered  as  practically  correct  for  all  desired  purposes  of  this  nature.  It  will  also 
be  shoAvn,  however,  that  when  it  comes  to  the  question  of  computations  involving  the 
actual  hitting  of  a  given  target,  there  are  forces  that  enter  to  divert  the  projectile 
from  the  original  vertical  plane  of  fire  enough  to  cause  it  to  fail  to  hit  the  point  aimed 
at  unless  allowance  be  made  for  them,  even  though  they  are  not  of  sufficient  moment 
to  introduce  serious  errors  into  computed  values  of  ranges,  angles,  times,  etc. 

EXAMPLES. 

Note  to  Examples  in  this  Book. — In  very  many  cases  one  example  gives  in 
tabular  form  the  data  for  a  considerable  number  of  separate  problems.  In  giving  out 
examples,  therefore,  give  one  problem  in  the  example  to  each  midshipman. 

1.  For  the  following  angles  of  elevation  and  jump,  what  are  the  corresponding 
angles  of  projection  and  departure,  the  position  angle  being  zero  ?  Draw  a  curve 
showing  each  an2:le. 


DATA. 

ANSWERS. 

Problem. 

Angle 
of  elevation. 

Angle  of 
jump. 

Angle  of  projection  = 
angle  of  departure. 

1 

3°     00' 
2       00 

2  15 

3  05 
3       35 
9       50 

+    5' 
+    3 
+    5 

—  5 
+    5 

—  10 

3°     05' 

2 

2       03 

3 

2       20 

4 

3       00 

5 

3       40 

6 

9       40 

26 


EXTERIOE  BALLISTICS 


2.  For  the  following  angles  of  elevation,  jump  and  position,  what  are  the  corre- 
sponding angles  of  projection  and  departure  ?    Draw  curves  showing  all  angles. 


DATA. 

ANSWERS. 

Problem. 

Angle 
of  elevation. 

Angle 
of  jump. 

Angle 
of  position. 

Angle 
of  departure. 

Angle 
of  projection. 

1 

2°     00' 
3       00 
3       00 

2  00 

3  00 

5  00 

4  00 

6  00 

+  5' 

—  3 

—  7 
+  4 
+  6 

—  5 
+  6 

—  8 

+  15°     00' 
+  12       15 
— 10       30 

—  12       07 
+  11       15 
+  10       16 

—  9       37 

—  6       22 

+ 17°     05' 
+  15       12 

—  7       37 

—  10       03 
+  14      21 
+  15       11 

—  5       31 

—  0      30 

+  2°     05' 

2 

+  2       57 

3 

+  2       53 

4 

+  2      04 

5 

+  3       06 

6 

+  4      55 

7 

+  4       06 

8 

+  5       52 

3.  A  target  is  at  a  horizontal  distance  of  3000  yards  from  the  gun,  and  is  750  feet 
higher  than  the  gun  above  the  water.  Compute  the  angle  of  position  by  the  use  of 
logarithms.  Answer.     /j  =  4°  45'  49". 

4.  A  target  is  at  a  horizontal  distance  of  10,000  yards  from  the  gun,  and  is  on 
the  water  1500  feet  below  the  level  of  the  gun,  the  latter  being  in  a  battery  on  a  hill. 
Compute  the  angle  of  position  by  the  use  of  logarithms. 

Anstver.     p={-)2°  51'  45". 

5.  A  target  is  at  a  horizontal  distance  of  1924  yards  from  the  gun,  and  is  1123 
feet  higher  above  the  water  than  the  gun.  Find,  by  the  use  of  the  traverse  tables,  the 
angle  of  position  and  the  distance  in  a  straight  line  from  the  gun  to  the  target  in 
yards.  Ansivers.     /?  =  11°   00'  00".    i2'  =  1960  yards. 

6.  A  target  is  at  a  horizontal  distance  of  1860  yards  from  the  gun,  and  it  is  on 
the  water  1238  feet  below  the  level  of  the  gun,  the  latter  being  in  a  battery  on  a  hill. 
Find,  by  the  use  of  the  traverse  tables,  the  angle  of  position  and  the  distance  in  a 
straight  line  from  the  gun  to  the  target  in  yards. 

Answers.     p={-)12°  30'  31".    i2'  =  1905.23  yards. 


CHAPTEE  3. 

THE  EQUATION  TO  THE  TRAJECTORY  IN  A  NON-RESISTING  MEDIUM  AND 
THE  THEORY  OF  THE  RIGIDITY  OF  THE  TRAJECTORY  IN  VACUUM. 

New  Symbols  Introduced. 

(.r,  y)  .  . . .  Coordinates  of  any  point  of  the  trajectory  in  feet, 
(x^^,  i/q)  .  .  .  .  Coordinates  of  the  highest  point,  or  vertex,  of  the  trajectory  in  feet. 
6.  . .  .  Angle  of  inclination  of  the  tangent  to  the  trajectory  at  any  point  to 

the  horizontal. 
i.  . .  .Elapsed  time  of  flight  from  the  muzzle  to  any  point  on  the  trajectory 

in  seconds. 
t^.  . .  .  Elapsed  time  to  the  vertex  of  the  trajectory  in  seconds. 
T .  . .  .  Time  of  flight  from  the  muzzle  to  the  point  of  fall  in  seconds. 
g.  . .  .Acceleration  due  to  gravity  in  foot-seconds  per  second;  g  =  32.2. 
d.v.  . .  .Differential  increment  in  x. 
dij .  .  .  .  Differential  increment  in  y. 
ds.  . .  .  Differential  increment  along  the  curve,  that  is,  in  s. 

32.  The  resistance  of  the  air  to  the  motion  of  a  projectile  animated  with  the  high 
velocity  given  by  a  modern  gun  is  so  great  that  calculations  which  neglect  it  are  of 
little  practical  value  at  the  present  day,  except  to  aid  in  a  comprehension  of  the 
underlying  principles  of  exterior  ballistics  and  to  permit  comparisons  to  be  made 
between  the  travel  of  a  projectile  in  a  non-resisting  and  in  a  resisting  medium.  For 
these  purposes,  however,  the  study  of  the  motion  of  a  projectile  in  vacuum  is  of  the 
utmost  value,  and  therefore  this  chapter  will  be  devoted  to  this  subject,  which, 
through  its  simplicity,  furnishes  a  valuable  groundwork  for  the  correct  understanding 
of  the  more  complex  problems  which  arise  when  account  is  taken  of  the  atmospheric 
resistance. 


— DL 


Figure  2, 


33.  Figure  2  represents  the  trajectory  in  vacuum,  the  origin,  0,  being  taken  at 
the  gun,  the  axis  of  Y  vertical,  and  the  axis  of  X  horizontal.  The  line  marked  V  is 
the  line  of  departure,  and  by  its  length  represents  the  initial  velocity,  V ;  the  vertical 


28  EXTEEIOE  BALLISTICS 

and  horizontal  components  of  which  are  V  sin  <f>  and  V  cos  <f),  respectively.  The 
remaining  velocity,  v,  at  any  point  of  the  trajectory,  P,  whose  coordinates  are  {x,  y), 
and  its  horizontal  component,  vn,  are  also  represented.  Letting  6  represent  the  angle 
at  which  the  tangent  to  the  trajectory  at  the  point  P  is  inclined  to  the  horizontal,  we 
have  Vh  =  v  cos  0. 

34.  Since  the  only  force  acting  on  the  projectile  after  it  leaves  the  gun  is  the 

vertical  force  of  gravity,  the  projectile  will  remain  throughout  its  travel  in  the 

vertical  plane  through  the  line  of  departure,  and  the  trajectory  will  be  a  plane  curve. 

Primary  35.  Let  t  be  the  elapsed  time  from  the  origin  to  any  point  P,  whose  coordinates 

egua  ions.  ^^^  ^^^  ^^  _  ^^^  ^^^^^  ^^^^^^  ^^^  figure  we  evidently  have : 

a;  =  ^ycos<^         y  =  tV  sm  4,  — Igt^  (1) 

as  ^gf^  represents  the  vertical  acceleration  (in  this  case  negative)  due  to  the  action 
Equation  to  of  the  forco  of  gravity  during  the  time  t.    Eliminating  t  between  the  two  equations 

trajectory   .      ,         , 

in  vacuum,  given  above  we  have 

y  =  xtain<f>—  .-.t/o^^  ■>  .  (^) 

^  2  y ^  cos-  (f> 

and  (2)  is  the  equation  to  the  trajectory  in  vacuum,  which  trajectory,  from  the  form 
of  its  equation,  is  evidently  a  parabola  with  a  vertical  axis. 

36.  From  the  above  equation  various  expressions  may  be  derived  from  which  we 
can  readily  determine  the  values  of  the  different  elements  of  the  curve. 

Angle  of  37.  Differentiatnig  (2)  and  putting  tan  (9  for  ^,  we  get 

inclination.  ^^ 

tan  6  =  tan  d)  — --„ -2— -—  (3) 

V'^  cos^  (j> 

which  gives  the  inclination  of  the  curve  to  the  horizontal  at  any  point. 
Horizontal      '     38.  Putting  ^  =  0  in  (2),  wc  find  two  values  of  x,  the  first  zero,  and  the  second 
range,    -r^o     •     -p  j, 

sm  /i(p    consequently  the  horizontal  range,  OH,  is  given  by 
9 

-^^  V^sm2<f>  .^. 

This  shows  that,  for  a  given  initial  velocity,  the  range  increases  with  the  angle  of 

V- 
departure  up  to  ^  =  45°,  when  it  reaches  its  maximum  value  of  — ;  and  that  the  same 

range  is  given  by  either  of  two  angles  of  departure,  one  as  much  greater  than  45°  as 
the  other  is  less  than  45°. 
Variations  39.  From  (3)  we  see  that,  as  x  increases,  d  decreases  from  its  initial  value  of  <^, 

in  angle  of  q^ 

inclination.   ^^^\^i\  j^  becomes  zero  when  tan  4>=  ,.„  ^   „  ,   :  that  is.  when 

V^  cos^  ^ 

_  72  cos^  </)  tan(/>  _  7^  cos^  4*  sin  "^  _  7^  sin  4  cos  4 
^~  g  ~        gcoscj>  g 


7^  sin  2<i  *                 V      V^  sin  2<^       ,  Z 

- — o„         ,  or,  as  X= -,  when  x  = 


2g         '  "^'  -  -  g        '  2 

We  also  see  that  after  this  value  of  x  is  reached,  the  value  of  0  becomes  negative,  as 

3^ then  becomes  greater  than  tan  4>;  and  that  for  x  =  X,  6=  —(f>.    That  is  to 

7^  cos-  <f> 

say,  the  highest  point  or  vertex,  S,  of  the  curve  is  midway  of  the  range,  and  the  angle 
of  fall,  w,  is  equal  to  the  angle  of  departure,  4. 


*  Sin  2(p  =  2  sin  ^  cos  <?>. 


GENERAL  AND  APPROXIMATE  DEDUCTIONS        29 

40.  The  maximum  ordinate  is  obtained  by  putting  x=j=        ^^        in  (3),  ^^"J^^t'e^*^^ 
whence  we  have  for  the  coordinates  (x^,  y^)  of  the  vertex 

_  V-  sin  24>  _X  _  Y^  sin-  (^  _  X  tan  <f>  /kx 

""^ 2g  3"  y^-        2g  4  ^""^ 

This  value  of  y^  is  directly  given  from  the  fact  that  the  vertical  component  of  the 
initial  velocity  is  V  sin  <^,  since  the  height  to  which  a  body  will  rise  in  vacuum  is 
equal  to  the  square  of  the  vertical  velocity  divided  by  2g. 

41.  The  vertical  and  horizontal  components  of  the  velocity  at  any  time  being  Remaining 

respectively  -^^  =  T'  sin  (f>  —  gt  and  -^  =V  cos  (f>,  we  have  for  the  remaining  velocity 
at  any  point  (x,  y) 

=  V'- - 2gtV  sin  <^  +  g't-  =  V- - 2g  (vt  sin  <p -  ^~\ 

whence  we  have 

v:=\/V'-2gy  (6) 

as  from  the  second  equation  of  (1)  we  have 

y=Vf  sin  cb-yt^ 
Thus  we  see  that  where  y  =  0,  both  at  the  origin  and  at  the  point  of  fall  on  the  hori- 
zontal plane,  v=V,  or  the  striking  velocity  is  the  same  as  the  initial  velocity.    Also, 

putting  y  =  y^  = — ^,  we  have  for  the  velocity  at  the  vertex  Vo  =  V  cos  <^,  which 

of  course  it  must  be,  since  the  vertical  component  of  the  velocity  has  been  destroyed  by 
gravitation,  while  the  original  horizontal  component  of  the  initial  velocity  remains 
unchanged  throughout  the  trajectory. 

42.  Since  the  horizontal  velocity  is  constant  and  equal  to  V  cos  cf),  it  is  evident  Time  of 

flight. 

that  the  time  to  any  point  whose  abscissa  is  x  is  given  by  ^=  = r  ,  and  so  the 

-^  ^  °  V  cos(f> 

duration  of  the  trajectory,  or  time  of  flight,  T,  is  given  by 

rp_      X      _     /2Z  tan  <f>  .r^\ 

Vcos<l>~  y        g  ^    ' 

The  second  of  these  values  is  obtained  directly  from  the  consideration  that  if  gravity 
did  not  act  the  projectile  would  rise  to  the  height  (X  tan  ^)  while  moving  X  hori- 
zontally, and  so  the  distance  which  it  falls  from  the  tangent  to  the  curve  at  the  origin 
under  the  action  of  gravity  during  the  time  of  flight  T  is  given  by 

X  tan  <i>  =  yT'' 

43.  To  determine  the  range,  X',  on  an  inclined  plane,  let  (x^,  y^)  be  the  coordi-   Range  on 
nates  of  the  target  M,  in  Figure  2,  p  being  the  angle  of  position,  and  i/^'  =  <^  — p  the 

angle  of  projection.    Then  from  (2)  we  get 


27- 
9 

sin  <f>cos  p  —  cps  (f)  sin  p  _  sin(<j!>  — ^) 


tan  o=  ^  =tan  <A—  ,^-,£^^  ^  a;.  =  cos"  ^(tan  <^  — tan  p) 

x^  ^     27^cos^<^  g 


But  tan  <i>  —  tan  p  — 

cos  <^  cos  p  cos  (f)  cos  p 

Therefore  x,=  ^-^  X  '^^i<l>-P)^0'<l>  =  ^ T^  x  s^^fcos(f  +  p) 

g  cos  p  g  cos  p 

Whence,  since  X'  —  x^  sec  p,  we  have 


X'=^^'  X  sinf  cos(f +  ;?)  /g) 

g  cos^  p 


*  Sin'  tt>  +  cos'  0  =  1. 


30 


EXTEEIOR  BALLISTICS 


dtpfrture  ^^-  "^^  determine  the  angle  of  departure  necessary  to  give  a  given  range  on  an 

on  an   inclined  plane,  we  have 

incline.  ^  ' 


But  x^  =  X'  COS  p  and  y^  =  X'  sin  p;  therefore 

X'  sin  p  =  X'  cos  p  tan  cA-  ?^^^^ 

SK^cos-t^ 

or  2X'V'^  cos  p  sin  <^  cos  <^  —  2X'y^  sin  p  cos^  </>  —  gX'^  cos-  p  =  0 

But  sin  ^  cos  0=-J  sin  3</>  and  cos-  ^  =  |(l  +  cos  2^),  therefore 

X'y^(sin  2^  cos  p  — cos  2^  sin  p)  =gX'^  cos^  p  +  Z'F^  sin  p 


whence 


sin(2(/)  — /))  =  ^^  cos^  p  +  sin  p 


(9) 


45.  If  p  were  zero,  as  the  angle  of  departure  was  \p',  we  would  have  for  the  hori- 

zontal  range  from  (8),  X= •  sin  i/^'  eosxp',  and  so  the  ratio  of  the  range  on  an 

inclined  plane  to  the  horizontal  range,  for  the  same  angle  of  projection,  would  be 

^=  ^^^^f'^f)=secp{l-tsind,'tsinp)  (10) 

X        cos  ij/  cos^  p 

The  value  of  the  second  member  of  (10)  is  very  nearly  unity  so  long  as  ij/'  and  p  are 
small  angles,  being,  for  example,  0.9992  for  i/^'  =  3°,  p  =  5° ;  and  it  therefore  follows 
that,  with  small  angles  of  projection  and  position,  we  may  consider  the  range  as 
independent  of  the  angle  of  position. 


Figure  3 


Theory  of  46.  The  assumption  in  the  last  line  of  the  preceding  paragraph  is  called  the 

of  the   assumption  of  the  rigidity  of  the  trajectory,  and  evidently  consists  in  supposing  that 

the  gun,  trajectory,  and  line  of  position  (a  chord  of  the  trajectory)  may  be  turned 

through  a  vertical  angle,  as  illustrated  in  Figure  3,  without  any  change  of  form.    The 

trajectory  is  assumed  to  be  rigid  in  practice  when  the  same  sight  graduations  are  used 


GENEEAL  AND  APPEOXIMATE  DEDUCTIONS        31 

in  firing  a  gun  at  objects  at  different  heights,  or  when  the  gun  itself  is  at  different 
heights  relative  to  the  target. 

47.  As  an  example  of  the  mathematical  work  relative  to  the  trajectory  in 
vacuum,  and  to  show  the  proper  logarithmic  forms  for  such  work,  suppose  we  have 
given,  in  vacuum,  an  angle  of  elevation  (i/')  of  7°  50',  an  angle  of  jump  (;)  of  +10', 
an  initial  velocity  of  2600  f.  s.,  and  an  angle  of  position  (p)  of  zero;  and  desire  to 
compute  the  horizontal  range,  time  of  flight,  striking  velocity,  angle  of  fall,  coordi- 
nates of  vertex,  and  time  to  and  remaining  velocity  at  the  vertex.  (Use  Table  VI 
where  convenient.) 

X  Z  tan  <i      ,        T        ,      -rr  ^^^  , 

3^0= -9-;  y^= — ^ '>  ^o=-2-;  i\=vco^<^ 

F  =  2600 2  log  6.82994 colog  6.58503  —  10 log  3.41497 

^  =  8°  00' sec  0.60425 tan  9.14780—10.  ..cos  9.99575  —  10 

20  =  16°  00'..  ..     sin  9.44034  — 10 

Sr=i32.2 colog  8.49214—10 

4 colog  9.39794—10 


JCi=  57865.5'...     log  4.76242 log  4.76242 log  4.76242 

7'  =  22.475 log  1.35170 


3/0  =  2033.1 log  3.30816  

v^  =  2574.65 log  3.41072 

Eesults. 

7?  =  19388.5  yards.  x^  =  9644.25  yards. 

r  =  22.475  seconds.  ^o  =  3033.1  feet. 

(0  =  8°  00'.  ^0  =  11.2375  seconds. 

i'^  =  2600f.  s.  -^o  =  2574.65  f.  s. 

48.  And,  again,  suppose  the  angle  of  position  (p)  to  be  30°,  the  angle  of  pro- 
jection (i{/')  to  be  2°  25'  45",  and  the  initial  velocity  2900  f.  s. ;  and  it  is  desired  to  find 
the  range  on  the  incline  and  the  time  of  flight,  both  in  vacuum. 

X'  =  ^^  sin  ^' cos (^'  +  /^)  .  CO         _^  .  therefore,  x=X'  cos  p,  and 
g  cos^  p  X 

t=^ ,  whence  T  =  ^'  ^^^  f  ,  where  cf>  =  il^'  +  p 

7  =  2900   2  log  6.92480 colog  6.53760-10 

p  =  30°  00'  00"    sec  0.06247 2  sec  0.12494 cos  9.93753-10 

f  =   2°  25'  45"    sin  8.62721-10 

<f>z=32°  25'  45"   cos  9.92637-10.  ...     sec  0.07363 

^  =  32.2   log  1.50786.  . .  .colog  8.49214-10 

2   log  0.30103 

Z'  =  24916.5    log  4.39649 log  4.39649 


r  =  8.8155    log  0.94525 

Eesults.     i2'  =  8305.5  yards.     T  =  8.8155  seconds. 


32 


EXTERIOE  BALLISTICS 


EXAMPLES. 

1.  If  the  initial  velocity  and  angle  of  departure  are  as  given  in  the  first  two 
columns  of  the  following  table,  compute  the  horizontal  and  vertical  components  of 
the  velocity  at  the  point  of  origin,  in  vacuum.  Give  results  obtained  by  both  the 
use  of  logarithms  and  by  the  use  of  the  traverse  tables  without  logarithms. 


DATA. 

ANSWERS. 

Problem. 

Initial 
velocity. 

f.  3. 

Angle 
of  departure. 

By  logs. 

By  traverse  tables. 

Vh- 

f.  s. 

f.  s. 

Vh- 

f.  s. 

f.  s. 

1       

1000 
1100 
1250 
1400 
1500 
1750 
2000 
2400 
2600 
2900 

2°  00'  00" 

3  15    42 

4  25    16 

5  10    25 
10     12    14 

7     30    00 

5     00    00 

9     21    15 

12     37    54 

17     24   24 

999 
1098 
1246 
1394 
1476 
1735 
1992 
2368 
2537 
2767 

35 
63 
96 
126 
266 
228 
174 
390 
569 
868 

999 
1098 
1246 
1394 
1476 
1735 
1992 
2368 
2537 
2767 

35 

2 

63 

3 

96 

4 

126 

5 

265 

6         

229 

7 

174 

8 

390 

9 

569 

10 

868 

Note. — It  will  be  seen  from  the  above  that  the  traverse  tables  give  the  results  correctly 
to  the  nearest  foot-second,  which  is  all  that  is  required  in  ordinary  work. 


2.  The  data  being  as  given  in  the  first  three  columns  of  the  following  table,  find 
the  results,  in  vacuum,  required  by  the  other  columns. 


DATA. 

ANSWERS. 

Problem. 

Initial 

velocity, 
f.  s. 

Angle 
of  eleva- 
tion. 

Angle 
of  jump. 

Angle 
of  depar- 
ture. 

Horizon- 
tal range. 
Yds. 

Time 

of  flight. 

Sees. 

Angle  of 
fall. 

Striking 
velocity. 

f.  s. 

1 

1000 
1100 
12.'^0 
1400 
1.500 
1750 
2000 
2400 
2600 
2900 

5°     27' 
4      32 
3       33 

2  15 

7  22 

8  12 
12       37 

7       50 

3  07 
16       35 

+   7' 
+    3 

—  3 

—  5 
+    6 

0 

—  7 

—  10 
+    3 
+    5 

5°     34' 
4      35 
3       30 

2  10 

7  28 

8  12 
12       30 

7       40 

3  10 
16       40 

1999 

1995 

1971 

1533 

6002 

8951 

17500 

15767 

7719 

47840 

6.03 

5.46 

4.74 

3.29 

12.11 

15.50 

26.89 

19.89 

8.92 

51.66 

5°     34' 
4      35 
3       30 

2  10 

7  28 

8  12 
12       30 

7       40 

3  10 
16      40 

1000 

2 

1100 

3 

1250 

4 

1400 

5 

1500 

6 

1750 

7 

2000 

8 

2400 

9        

2600 

J      10 

2900 

GENERAL  AND  APPROXIMATE  DEDUCTIONS 


33 


3.  Given  the  initial  velocities  and  angles  of  departure  in  the  table  below,  com- 
pute the  coordinates  of  the  vertex,  and  the  time  to  and  remaining  velocity  at  the 
vertex,  in  vacuum.    . 


DATA. 

ANSWERS. 

Problem. 

Initial 
velocitv. 

f.  s.' 

Angle 
of  departure. 

Yds. 

Feet. 

to. 
Sees. 

f.  s. 

1 

1000 
1100 
1250 
1400 
1500 
1750 
2000 
2400 
2600 
2900 

5°  34' 
4   35 
3   30 

2  10 

7  28 

8  12 
12   30 

7   40 

3  10 
16   40 

999 

998 

986 

767 

3001 

4475 

8750 

7884 

3860 

23920 

146 

120 

90 

44 

590 

967 

2910 

1592 

320 

10742 

3.01 
2.73 
2.37 
1.64 
6.05 
7.75 

13.44 
9.94 
4.46 

25.83 

995 

2 

1096 

3  

1248 

4 

1399 

5 

1487 

6 

1732 

7 

1953 

8 

2379 

9 

2596 

10 

2778 

4.  A  body  is  projected  in  vacuum  with  r=1000  f.  s.,  and  an  angle  of  departure 
of  30°.    Where  is  it  after  3  seconds  ?    Where  after  10  seconds  ? 

Ansivers.     3  seconds.     a;=:1500V3  feet.     t/  =  1355  feet. 
10  seconds.     a;=5000V3  feet.     tj  =  3390  feet. 

5.  A  body  is  projected  in  vacuum  from  the  top  of  a  tower  200  feet  high,  with  a 
velocity  of  50  f.  s.,  and  an  angle  of  departure  of  60°.  Find  the  range  on  the  hori- 
zontal plane  through  the  foot  of  the  tower,  and  the  time  of  flight. 

Answers.     Range  =  128  feet.    Time  =  5.13  seconds. 

6.  What  is  the  angle  of  departure  in  vacuum  in  order  that  the  horizontal  range 
may  be:  (a)  Equal  to  the  maximum  ordinate  of  the  trajectory;  and  (b),  equal  to 
three  times  the  maximum  ordinate  ? 

Answers,     (a)   75°  57' 51".    (b)   53°  07' 48". 

^  7.  Compute  the  initial  velocity  and  angle  of  departure  in  vacuum  in  order  that 

the  projectile  may  be  100  feet  high  at  a  horizontal  distance  from  the  gun  of  a  quarter 
of  a  mile,  and  may  have  a  horizontal  range  of  one  mile. 

Answers.     For  1  mile  =  5280  feet.     5°  46' 05".       922  f .  s. 
For  1  mile  =  6080  feet.     5°  00' 07".     1062  f.  s. 

8.  The  angle  of  position  is  45°,  the  angle  of  projection  is  1°  16'  31",  and  the 
initial  velocity  is  1500  f.  s.  Compute  the  range  on  the  incline  and  the  time  of  flight, 
in  vacuum.  Answers.     i?'=:  1433  yards.     7  =  2.93  seconds. 

9.  What  must  be  the  angle  of  projection  in  vacuum  for  an  initial  velocity  of 
400  f.  s.  in  order  that  the  range  may  be  2500  yards  on  a  plane  that  descends  at  an 
angle  of  30°  ? 

Answer.     Angle  of  projection.     34°  35'  56"  or  85°  24'  04". 

10.  A  body  is  projected  in  vacuum  with  an  angle  of  departure  of  60°,  and  an 
initial  velocity  of  150  f.  s.  Compute  the  coordinates  of  its  position  and  its  remaining 
velocity  after  5  seconds ;  also  the  direction  of  its  motion. 

Atiswers.     a-  =  375feet.    7/  =  247feet. 

^=(-)22°  31'25".    i;  =  81f.  s. 


34  EXTERIOE  BALLISTICS 

11.  A  12"  mortar  shell  weighing  610  pounds,  fired  with  an  initial  velocity  of 
591  f.  s.,  and  an  angle  of  departure  of  73°,  gave  an  observed  horizontal  range  in  air 
of  1939  yards,  and  a  time  of  flight  of  36  seconds.  What  would  the  range  and  time  of 
flight  have  been  in  vacuum  ?         Ansivers.     5  =  2022  yards.     r  =  35.10  seconds. 

12.  The  measured  range  in  air  of  a  12"  shell  of  850  pounds  weight,  fired  with 
2800  f.  s.  initial  velocity,  and  an  angle  of  departure  of  7°  32',  was  11,900  yards,  and 
the  time  of  flight  was  19.5  seconds.  What  would  the  range  and  time  of  flight  have 
been  in  vacuum ?  Ansivers.     i2=: 21,097  yards.    r  =  22.8  seconds. 


CHAPTEE  3. 


THE  RESISTANCE  OF  THE  AIR,  THE  RETARDATION  RESULTING  THERE- 
FROM, AND  THE  BALLISTIC  COEFFICIENT. 


w. 

d. 

a. 

A. 

R. 

Rf. 

Rs' 

8,. 

8. 


c. 
C. 

/• 
/?■ 
Y. 

dv. 
dt. 

K. 


New  Symbols  Introduced. 

.  .  Weight  of  projectile  in  pounds. 

.  .  Diameter  of  projectile  in  inches. 

. .  Mayevski's  exponent, 

. .  Mayevski's  constant. 

. .  Total  air  resistance  in  pounds. 

.  .Total  air  resistance  under  firing  conditions  in  pounds. 

. .  Total  air  resistance  under  standard  conditions  in  pounds. 

.  .  Standard  density  of  air,  taken  as  unity. 

. .  Density  of  air  at  time  of  firing,  and  subsequently  representing  the 

,.8        8 
ratio  Y"  —  'i"  • 

.  ,  Coefficient  of  form  of  the  projectile. 

.  .  Ballistic  coefficient. 

, .  Altitude  factor. 

.  .  Integration  factor. 

.  .Maximum  ordinate,  or  ordinate  of  the  vertex,  in  feet;  Y  =  yQ. 

.  .  Differential  increment  in  v. 

. .  Difi^erential  increment  in  t. 

. .  Constant  part  of  ballistic  coefficient  for  a  given  projectile;  K=~-j^, 


49.  The  first  investigations  of  the  resistance  offered  by  the  air  to  and  the 
resultant  retardation  in  the  travel  of  the  projectile  were  in  the  nature  of  practical 
experiments  conducted  from  time  to  time  by  a  number  of  persons,  and  indeed, 
although  later  mathematical  investigations  have  led  to  a  fuller  understanding  of  the 
subject,  the  formulas  still  in  use  for  the  determination  of  atmospheric  resistance  and 
tlie  retardation  resulting  from  it  are  the  results  of  these  experiments ;  in  other  words, 
they  are  partly  empirical,  and  not  purely  mathematical.  The  list  of  men  who  con- 
ducted these  experiments  includes  many  names  prominent  in  the  records  of  scientific 
research,  such  as  Tartaglia,  Galileo,  Newton,  Bernouilli,  Eobins,  Count  Eumford, 
Dr.  Hutton,  Wheatstone,  Bashforth,  Mayevski  and  Zaboudski.  The  results  of  Bash- 
forth's  experiments,  as  expressed  in  formulas  by  Mayevski,  and  modified  and  extended 
by  Zaboudski,  form  the  basis  upon  which  calculations  for  resistance  and  retardation 
still  rest. 

50.  As  it  is  manifestly  simpler  to  determine  experimentally  the  retardation  pro- 
duced in  the  flight  of  a  projectile  than  it  is  to  attempt  to  measure  the  atmospheric 
pressure  opposing  its  motion,  the  experiments  have  naturally  taken  that  direction. 
To  measure  the  retardation,  a  gun  is  given  a  very  slight  elevation,  and  the  velocity 
of  the  projectile  is  measured  at  two  points  sufficiently  far  apart  to  make  the  two 
velocities  (v^  and  v^)  appreciably  different,  and  yet  near  enough  together  to  ensure, 
as  closely  as  possible,  that  the  resistance  of  the  air  does  not  change  from  one  point  of 
measurement  to  the  other.  The  two  pairs  of  screens  used  for  measuring  the  velocities 
should  be  at  about  the  same  level,  so  that  the  effect  of  gravity  may  be  neglected. 


Measure- 
ment of 
retardation, 


36  EXTEEIOR  BALLISTICS 

Having  determined  these  two  velocities,  it  is  evident  that  the  retardation  of  the  pro- 
jectile while  traveling  the  distance,  x,  between  the  two  points  of  measurement  is 
Vj^  —  v^.  Let  w  he  the  weight  of  the  projectile,  R  the  resistance  of  the  air  in  pounds 
(total  resistance).  Then,  since  the  work  done  by  the  resistance  must  equal  the  loss 
of  energy  of  the  projectile,  we  have 


whence. 


K=^(V-«.')  (11) 

where  R  is  taken  to  be  the  total  resistance  of  the  air  in  pounds  which  corresponds  to 
the  mean  velocity  — ^ — ~  . 

As  an  example  of  the  use  of  the  above  formula,  suppose  we  have  a  12"  pro- 
jectile weighing  870  pounds,  fired  through  two  pairs  of  screens  300  feet  apart,  and 
the  measured  velocity  at  the  first  pair  was  2819  f.  s.  and  at  the  second  pair  was 
2757  f.  s.  These  measured  velocities  are  the  mean  velocities  for  the  spaces  traversed 
between  the  two  screens  of  each  pair,  that  is,  we  may  assume  that  each  velocity  is  the 
velocity  at  the  point  midway  between  the  two  screens  of  its  own  pair.  The  distance 
given  as  300  feet  is  the  distance  between  the  midway  points  of  each  pair  of  screens. 

Also,  for  determining  the  value  of  iVj^  —  V2^),  we  know  that  v-^^  —  v^^^  (v^  +  Vo) 
(I'j  — tjg),  and  the  work  becomes: 

v^  +  V2  =  557G    log  3.74632 

v^-V2  =  62    log  1.79239 

w  =  870    log  2.93952 

a:=300    log  2.47712 colog  7.52288-10 

2^  =  64.4    log  1.80889 colog  8.19111-10 

5  =  15567.5  pounds log  4.19222 

in  which  R  is  the  resistance  for  the  mean  of  the  two  measured  velocities,  that  is,  for 

^1  +  ^2-2788  18. 

Expert-  51.  As  the  result  of  many  such  measurements  with  different  projectiles  and 

results,   different  velocities,  it  has  been  shown  that  the  resistance  of  the  air  is  proportional  to  : 

1.  The  cross-sectional  area  of  the  projectile;  or,  what  is  the  same  thing,  the 
square  of  its  diameter,  which  is  the  caliber. 

2.  The  density  of  the  air;  or,  what  is  the  same  thing,  the  weight  of  a  cubic  foot 
of  the  air. 

3.  A  power  of  the  velocity,  of  which  the  exponent  varies  with  the  velocity,  but 
may  be  considered  as  a  constant  within  certain  limits  of  velocity. 

4.  A  coefficient  which  varies  with  the  velocity,  with  the  form  of  the  projectile, 
and  with  the  assumed  value  of  the  exponent;  but  which  may  be  considered  as  a  con- 
stant for  any  given  projectile  between  the  same  limits  of  velocity  for  which  the 
exponent  is  considered  as  a  constant. 

Mayevski's  52.  In  accordance  with  these  four  experimentally  determined  laws,  we  may  write 

a  general  formula  expressing  the  retardation  of  a  projectile  caused  by  the  atmospheric 
resistance  to  its  flight ;  which  is  Mayevski's  formula.    It  would  be : 

Retardation  =  f^-  =  -A  ^^  i'«  ( 12) 

dt  w 

in  which  a  is  Mayevski's  exponent,  A  is  Mayevski's  constant  coefficient,  c  is  the 


formula, 


GENEEAL  AND  APPROXIMATE  DEDUCTIONS 


37 


coefficient  of  form  of  the  projectile,  d  the  diameter  of  the  projectile,  w  the  weight  of 
the  projectile,  and  v  the  velocity.    The  acceleration,  which  is  negative  in  this  case, 

is  of  course  represented  by  -  ^ . 

53.  The  quantity  S  in  the  above  equation  represents  the  ratio  of  the  density  of 
half-saturated  air  for  the  temperature  of  the  air  and  barometric  height  at  the  time  of 
firing  to  the  density  of  half-saturated  air  for  15°  C.  (59°  F.)  and  750  mm.  (39.5275") 
barometric  height.  The  values  of  8  for  different  readings  of  the  barometer  (in  inches) 
and  thermometer  (in  degrees  Fahrenheit)  may  be  found  in  Table  III  of  the  Ballistic 
Tables. 

54.  In  the  above  expression,  c  is  the  coefficient  of  form  of  the  projectile.  It 
will  be  readily  understood  that  if  certain  results  are  obtained  with  a  projectile  of  a 
given  shape,  a  change  in  the  shape  of  the  projectile  will  change  the  results.  There- 
fore the  factor  c  is  introduced,  and  values  for  it  for  different  projectiles  are  deter- 
mined experimentally,  as  explained  later. 

55.  Mayevski  adopted  as  standard  the  form  of  projectile  in  most  common  use  at 
the  time  he  conducted  his  experiments,  which  was  one  about  three  calibers  in  length, 
with  an  ogival  head  the  radius  to  the  curve  of  which  ogive  was  two  calibers,  and  for 
that  projectile  called  the  value  of  c  unity.  He  also  used  a  temperature  of  59°  F.  and 
a  barometric  height  of  29.5275"  as  standard,  thus  reducing  the  value  of  8  to  unity 
also.    His  general  expression  then  becomes 

dv 

m 

By  determining  velocities  experimentally  as  explained  in  paragraph  50,  he  proceeded, 
on  this  formula  as  a  basis,  to  derive  specific  laws  for  finding  the  retardation  at 
difl^erent  velocities. 

56.  As  the  result  of  these  experiments  he  derived  the  following  expressions : 


w 


(13) 


V  between  3600  f. 

s.  and  2600  f.  s. 

dv           .    d- 

~dr~    ^^  w 

^1.55 

log  li  =  7.60905- 

-10 

V  between  2600  f. 

s.  and  1800  f.  s. 

dv            .     d'^ 
dt                 w 

^1.7 

log  ^2  =  7.09620- 

-10 

V  between  1800  f. 

s.  and  1370  f.  s. 

dv  _      A     d- 
dt   ~        ^  w 

^2 

log  ^3  =  6.11926- 

-10 

V  between  1370  f. 

s.  and  1230  f.  s. 

dv            .     d- 
dt    ~     ^*   w 

^3 

log  .1,  =  2.98090- 

-10 

V  between  1230  f. 

s.  and  970  f.  s. 

dv            ^     d- 
dt   ~        '  w 

j;5 

log  15  =  6.80187- 

-20 

V  between  970  f.  s 

and  790  f.  s. 

dv            .     d^ 

dt     ~           «    IV 

^3 

log  .46  =  2.77344- 

-10 

V  between  790  f .  s 

.  and  0  f .  s. 

dv  _       A     d- 
dt    ~     ^  '  w 

y2 

log  1,  =  5.66989 - 

-10 

(14) 


38  EXTERIOR  BALLISTICS 

57.  From  the  above  expressions,  by  using  the  appropriate  one,  the  retardation 
for  any  velocity  in  foot-seconds  may  be  calculated  for  the  standard  projectile  and 
standard  condition  of  atmosphere  as  adopted  by  Mayevski ;  and  thence  for  any  other 
projectile  or  atmospheric  condition  by  applying  the  proper  multipliers.  Of  course  the 
total  resistance  of  the  air  in  pounds  (R)  may  be  found  by  multiplying  the  mass  of  the 
projectile  by  the  retardation,  so  we  have 

R  =  A  —  v<'X~=A-^^~v^  (15) 

w  g  g 

58.  For  instance,  given  a  6"  shell  weighing  105  pounds,  traveling  with  a  velocity 
of  2500  f.  s. ;  to  find  the  resistance  and  retardation  under  standard  atmospheric 
conditions,  provided  it  be  a  standard  shell. 

dv  A  d^     a         -D      A  d""     „ 

-^rr  =  —A  —  V"         R  =  A  —  v" 
at  w  g 

For  2500  f.  s.,  Mayevski's  constants  are  a=1.7  and  log  A  =  7.09620 -10. 

t'  =  2500    log  3.39794 loglog  0.53121 

a  =  1.7    log  0.23045 

t'"=    log  5.77640 ...loglog  0.76166 

A=    log  7.09620-10 

^'  =  36    loo-  1.55630 


^(fV    log  4.42890 log  4.42890 

w  =  105    log  2.02119 

g  =  32.2    log  1.50786 

-—  =-255.69  f.  s log  2.40771 


i2  =  833.75  pounds   log  2.92104 

In  the  ease  of  the  experimental  firing,  suppose  the  above  shell  gave  measured  velocities 
of  2525  f .  s.  and  2475  f .  s.  at  two  points  488.88  feet  apart,  to  find  the  resistance : 

^=  Yfx  ("^'-^">  =  Jgx^"^  +  '^^  ("^-"^) 

Vj  +  V2  =  5000    log  3.69897 

Vj-t;,  =  50    log  1.69897 

w  =  105 log  2.02119 

2^  =  64.4    log  1.80889 colog  8.19111-10 

a;  =  488.88    log:  2.68920 colog  7.31080-10 


22  =  833.75  pounds log  2.92104 

59.  These  results  being  only  for  standard  conditions,  we  must  introduce  another 
factor  if  we  desire  results  for  any  other  conditions.  This  factor  is  known  as  the 
ballistic  coefficient,  and  is  denoted  by  C.  It  is  a  most  important  quantity  to  which 
great  attention  must  be  paid  and  which  we  must  strive  to  thoroughly  understand,  for 
it  enters  constantly  into  nearly  every  problem  in  exterior  ballistics.  It  represents 
the  combination  of  the  different  elements  already  explained  as  well  as  some  other 
elements  which  will  now  be  discussed. 

60.  Introducing  the  ballistic  coefficient,  equation  (12)  becomes: 

—  =-—«;«  a6) 

dt  C  ^     ' 

In  other  words,  the  values  resulting  from  the  use  of  the  specific  formulae  given  in  (14) 
must  be  divided  by  C  for  the  individual  case  and  conditions  in  order  to  get  results 
for  any  other  than  standard  conditions. 


GENERAL  AND  APPROXIMATE  DEDUCTIONS 


39 


61.  The  value  of  C  in  its  most  complete  form  is  given  by  the  expression : 


Ballistic 
coefficient. 


in  which 


r-  iii^ 


(17) 


«'  =  the  weight  of  the  projectile  in  pounds. 
d  =  the  diameter  of  the  projeetile  in  inches  (caliber  of  the  gun). 
c  =  the  coefficient  of  form  of  the  projectile. 
/  =  the  altitude  factor. 
;8  =  the  integration  factor. 
8  =  the  density  of  the  air  at  the  time  of  firing. 
Si  =  the  adopted  standard  density  of  the  air. 

62.  Considering  these  factors  in  detail : 

IV  and  d  are  standard  characteristics  of  the  projectile,  and  need  no  special  remark. 

c  is  a  quantity  representing  the  form  of  the  projectile.  It  may  readily  be  con- 
ceived that  for  a  projectile  more  sharply  pointed  than  the  standard  its  value  would 
be  less  than  unity  and  that  the  shell  would  suffer  less  retardation  than  the  standard 
projectile  (for  the  modern  12",  long-pointed  shell,  for  instance,  c  =  O.Gl),  whereas 
for  a  blunter  shell,  its  value  would  exceed  unity  (for  a  flat-headed  projectile  the 
value  of  c  is  probably  greater  than  3) .  It  has  been  found  that  the  value  of  c  depends 
to  some  extent  upon  the  smoothness  and  the  length  of  the  projectile,  but  that  it 
depends  primarily  upon  the  shape  of  the  head.  Apparently  the  form  of  the  head 
near  its  junction  with  the  cylindrical  body  is  also  a  most  important'  factor  in  deter- 
mining the  resistance  of  the  air  to  the  flight  of  the  projectile.  The  U.  S.  Navy 
method  of  determining  the  value  of  c  for  any  projectile  by  experimental  firing  will 
be  explained  later. 

§1  represents  the  adopted  standard  density  of  the  air,  which  has  been  taken  as 
the  density  of  half-saturated  air  for  15°  C.  (59°  F.)  and  750  mm.  (29.5275") 
barometric  height.  8  represents  the  density  of  half-saturated  air  for  the  given  tem- 
perature and  barometric  height  at  the  time  of  firing.  Assuming  that  8^  =  1  for  the 
standard  condition,  Table  III  of  the  Ballistic  Tables  has  been  computed  for  the 

values  of  -s—  for  different  readings  of  the  thermometer  in  degrees  Fahrenheit  and 
^1 

8  8  8 

barometer  in  inches;  and  as  -^^then  becomes ^r-,  when  this  table  is  used,  -«— may  be 

f>i  1  Ol 

replaced  by  8  in  the  formulae.    Hereafter  the  symbol  8  will  be  used  to  represent  this 

TW 

ratio,  and  (17)  then  becomes  C'  =  ^^— ,-,  in  which  8  is  taken  from  Table  III. 

/  is  a  factor  which  enters  in  cases  in  which  we  can  no  longer  assume  that  the 
density  of  the  air  is  the  same  at  all  points  of  the  trajectory,  owing  to  the  fact  that 
the  path  of  the  projectile  rises  to  a  considerable  height  above  the  level  of  the  gun,  or 
passes  to  a  considerable  vertical  distance  below  the  level  of  the  gun,  as  when  firing 
from  an  elevated  battery  down  to  the  water.  In  such  cases,  /,  which  is  known  as  the 
altitude  factor,  must  enter  into  the  computation  of  the  value  of  the  ballistic  coefficient. 
/  is  the  ratio  of  the  density  of  the  air  at  the  gun  to  the  mean  density  of  the  air  through 
which  the  projectile  actually  passes.  The  mean  height  of  the  projectile  during  flight 
is  ordinarily  taken  as  two-thirds  the  height  of  the  vertex  of  the  trajectory,  which 
would  be  exact  were  the  trajectory  in  air  a  true  parabola.  This  is  therefore  ordinarily 
practically  correct,  and  the  mean  density  of  the  air  is  taken  to  be  the  same  as  the 
density  at  a  height  of  |1'.  The  value  of  /  for  any  height  in  feet  will  be  found  in 
Table  V  of  the  Ballistic  Tables. 


Weight  and 
diameter. 

Coefficient 
of  form. 


Density 
factor. 


Altitude 
factor. 


40 


EXTERIOR  BALLISTICS 


For  firing  when  gun  and  target  are  in  approximately  the  same  horizontal  plane, 
correction  for  altitude  is  ordinarily  a  needless  refinement  for  trajectories  for  which 
the  time  of  flight  does  not  exceed  about  12  seconds.  In  computing  range  table  data, 
the  correction  for  altitude  is  generally  started  when  such  correction  would  produce  a 
variation  in  the  angle  of  departure  of  about  one  minute  in  arc. 

In  all  problems  in  which  the  vertical  distance  of  the  point  aimed  at  above  or  below 
the  horizontal  plane  of  the  gun  is  such  that  the  rigidity  of  the  trajectory  cannot  be 


Figure 


taken  for  granted,  the  ballistic  coefficient  should  be  corrected  for  altitude;  and  an 
examination  of  Figure  4  will  show  that  the  mean  height  of  the  trajectory  from  gun 
to  target  is  less  or  greater  than  two-thirds  the  height  of  the  target  above  the  hori- 
zontal plane  of  the  gun  according  as  the  target  is  nearer  to  or  further  from  the  gun 
than  the  vertex  of  the  complete  trajectory.    Thus,  if  the  target  be  at  A^,  the  mean 

height  of  the  arc  OA^  is  less  than  f^/^,  and  approaches  -~-  more  and  more  as  A-^  is 

nearer  and  nearer  0;  while  if  the  target  be  at  Ao  the  mean  height  of  the  arc  OA2  is 
greater  than  ^y^,  being  about  equal  to  2/2  when  the  abscissa  of  A,  is  fX.  If,  therefore, 
all  possible  refinements  are  to  be  introduced  into  the  calculations,  the  relative  positions 
of  target  and  vertex  must  be  determined  before  fixing  the  value  of  /.  Generally  speak- 
ing, however,  it  will  be  sufficiently  accurate  to  give  /  the  value  corresponding  to 


.  .Y-Y. 


FlGUEE    5. 


The  two  preceding  paragraphs  explain  how  to  determine  the  value  of  /  for  either : 
(1)  A  horizontal  trajectory  with  a  maximum  ordinate  sufficiently  great  to  make  it 
necessary  to  correct  for  altitude;  or  (2)  in  the  case  in  which  there  is  a  material 
difference  in  height  between  the  gun  and  the  target.  For  the  third  possible  case, 
a  very  likely  one  in  naval  operations,  that  of  long-range  firing  at  an  elevated  target, 
it  is  apparent  that  we  have  here  both  of  the  conditions  calling  for  the  use  of  the  factor 
/  as  previously  discussed.  In  the  light  of  what  has  been  said,  an  approximate  rule 
that  would  probably  not  lead  to  material  error  in  most  cases,  is  to  take  the  value  of  / 


GENERAL  AND  APPROXIMATE  DEDUCTIONS        41 

from  the  table  for  a  height  equal  to  two-thirds  the  maximum  ordinate  or  two-thirds 
the  height  of  the  target,  whichever  of  the  two  gives  the  greater  height. 

The  rule  given  in  the  preceding  subparagraph  is  of  course  only  approximate,  and 
a  reference  to  Figure  5  will  show  at  once  that  a  closer  approximation  to  the  mean 
height  of  travel  would  really  be  ^a  +  ^Y  sec  p,  and  that  the  value  of  /  would  then  be 
taken  from  the  table  for  the  height  determined  by  the  above  expression  (not  for  two- 
thirds  of  it) .  It  is  believed,  however,  that  the  first  rule  given  is  sufficiently  accurate 
for  all  ordinary  cases;  although  special  consideration  should  be  given  to  this  point 
in  all  cases  involving  peculiar  conditions. 

^  is  a  quantity  known  as  the  integration  factor,  and  will  be  explained  later.    For  integration 
the  present  it  may  be  assumed  to  be  equal  to  unity,  and  it  will  therefore  disappear 
from  the  formula  for  the  value  of  the  ballistic  coefficient  for  all  our  practical  purposes. 

63.  C,  as  given  in  its  fullest  form  in  paragraph  61,  is  sometimes  known  as  the 

reduced  ballistic  coefficient,  the  form  used  by  Mayevski,  C  =  — ,^ ,  being  called  the 

ballistic  coefficient.  The  expression  for  the  value  of  this  ballistic  coefficient  should 
always  be  remembered  in  its  fullest  form,  however,  and  the  different  factors  entering 
into  it  allowed  to  drop  out  by  becoming  unity  as  the  conditions  of  actual  firing 
approach  the  standard  conditions. 

64.  Suppose  we  desire  to  find  the  value  of  the  ballistic  coefficient  for  the  12"  gun, 
w  =  870,  c  =  0.61,  for  30.1-1"  barometer  and  24.5°  F.,  when  the  highest  point  of  the 

trajectory  is  3333  feet.     The  formula  is  C  =  -^,^.    We  could  work  it  out  directly 

from  this  formula,  but  where  investigations  are  to  be  carried  out  in  regard  to  any  one 
particular  gun  and  projectile,  it  is  convenient  to  work  out  the  combined  value  of  the 
constant  factors  for  that  projectile,  that  is,  for  iv,  c  and  dr,  and  having  once  determined 
this,  thereafter  for  the  given  gun  and  projectile  we  have  only  to  apply  8  and  /  to  this 
constant  to  get  the  value  of  the  ballistic  coefficient  under  the  given  conditions. 
Expressed  mathematically  this  is : 

K=^         C=J-K 
cd-  8 

and  for  the  above  problem  the  work  becomes : 

w  =  8r0    log  2.93953 

c  =  .61    log  9.78533-10 colog  0.21467 

d^  =  14:-i    loir  2.15836 colog  7.84164-10 


Z  =  9.9045    log  0.99583 

Now,  from  Table  III,  for  30.14"  and  24.5°,  we  find  8=  1.0947.    And,  from  Table  V, 

for  a  height  of =2222  feet,  we  find  /=  1.059.    Hence,  for  our  special  case, 

o 

we  have 

A'=    log  0.99583 

/=  1.059 log  0.02490 

8  =  1.0947    log  0.03929 coloir  9.96071-10 


(7  =  9.5816    log  0.98144 

65.  Referring  to  the  second  part  of  paragraph  58,  we  see  that  we  found  experi- 
mentally that  a  certain  resistance  existed  at  the  time  of  firing  to  the  passage  of  a 
certain  projectile  through  the  air.  Let  us  now  suppose  that  the  coefficient  of  form  of 
the  projectile  used  was  c=0.61,  and  that  the  barometer  stood  at  30.14",  and  the 
thermometer  at  24.5°  F.  at  the  time  of  firing.    What  would  be,  from  this  experimental 


42  EXTEEIOR  BALLISTICS 

firing,  the  resistance  to  a  standard  projectile  under  standard  atmospheric  conditions? 
The  air  on  firing,  being  more  dense  than  standard,  the  resistance  would  be  less  under 

standard  conditions,  that  is,  Rs  =  -„-  Rf.    The  projectile  used  being  more  tapering  than 

o 

the  standard,  would  pass  more  easily  through  the  air,  and  the  resistance  to  the 
standard  projectile  would  be  more  than  that  measured,  that  is,  jRg=  — Rfj  and  com- 
bining, Rs=j-Rf 

Rf  =  833.7b    log  2.92104 

8=1.0947    log  0.03929 colog  9.96071-10 

c  =  .61    log  9.78533-10 colog  0.21467 

i2,  =  1248.6  pounds log  3.09642 

66.  Suppose  that  we  have  given  that,  for  a  12"  gun:  w  =  870;  c  =  0.61;  two 
measured  velocities  at  points  920  feet  apart  were  2840  f .  s,  and  2810  f .  s. ;  to  deter- 
mine the  resistance,  and  then  discuss  the  difl:erence  between  the  results  obtained  by 
actual  firing  and  by  the  use  of  Mayevski's  formula.  For  simplicity  in  computation, 
consider  the  atmospheric  conditions  as  standard.    By  Mayevski's  formula 

R  =  A^^  v'';    Mean  velocity  =  2825  f.  s.;    a  =  1.55;    log  A  =  7.60905 -10 

i;  =  2825    log  3.45102 loglog  0.53794 

a=1.55    log  0.19033 

v«=    log  5.34900 loglog  0.72827 

c=.61    log  9.78533-10 

A=    log  7.60905-10 

d-  =  lU    log  2.15836 

g  =  32.2    log  1.50786 colog  8.49214-10 

B  =  2476.7  pounds log  3.39388 

By  actual  firing :    R=  ^''_  {v^--v^-)=  -^(v^  +  v.)  (v^-v^) 

t;,  +  r,  =  5650    log  3.75205 

t;i-V2  =  30    log  1.47712 

w  =  870    log  2.93952 

2g  =  64:A    log  1.80889 colog  8.19111-10 

a;  =  920    log  2.96379 colog  7.03621-10 

72  =  2488.9  pounds log  3.39601 

If  by  our  experimental  firing  we  find  as  above  that  the  resistance  is  2488.9  pounds 
for  the  given  projectile,  when  moving  with  a  velocity  of  2825  f .  s.,  and  assuming  that 
at  this  velocity  the  resistance  varies  as  the  1.55th  power  of  the  velocity,  what  would 
be  the  value  of  Mayevski's  constant  A  in  this  case? 

E^A  —  V      therefore      A  =  -,?—  R 
g  cd-V^ 

72  =  2488.9    log  3.39601 

v^  (from  preceding  problem)  .  .  .log  5.34900 colog  4.65100  —  10 

^  =  32.2    log  1.50786 

c=0.61    log  9.78533-10 colog  0.21467 

^2  =  144    log  2.15836 colog  7.84164-10 

A  =  .0040849 log  7.61118-10 


GENERAL  AND  APPROXIMATE  DEDUCTIONS 


43 


But  Mayevski  gives  for  A  for  this  velocity  a  value  of  A  =  .0040649,  of  which 
log  A  =  7.60905 -10. 

This  difference  simply  means  that  Mayevski's  value  is  the  mean  of  a  large  num- 
ber of  values  obtained  by  experimental  firings,  such  as  the  one  worked  out  above ;  and 
that  the  value  of  A  found  above  is  the  one  resulting  from  a  single  firing  only.  The 
difference  between  them  therefore  simply  represents  the  difference  between  a  mean 
value  resulting  from  many  experiments  and  one  of  the  many  individual  values  that 
go  to  make  up  such  a  mean. 

Note. — See  Chapter  15  for  a  more  full  discussion  of  the  coefficient  of  form,  the  subject 
being  there  treated  according  to  the  most  modern  practical  methods  now  employed  in  the 
U.  S.  Navy.  In  this  later  consideration  a  different  method  of  determining  the  values  of  the 
coefficient  of  form  and  ballistic  coefficient  is  employed,  and  in  fact  a  somewhat  different 
conception  of  what  the  coefficient  of  form  really  is  is  adopted.  A  comprehension  of  the 
methods  of  this  present  chapter  is  however  very  necessary  in  understanding  the  explana- 
tions contained  in  Chapter  15. 


EXAMPLES. 

1.  Compute  the  value  of  K  ic=^  J-^  =  "{^-^j  i^  the  ballistic  coefficient  in  the 

cases  given  in  the  following  table ;  giving  the  values  of  log  K  and  colog  E.  Do  not 
use  Table  VI  of  the  Ballistic  Tables  in  these  computations;  as  this  question  calls  for 
the  computation  of  the  data  contained  in  that  table. 


DATA. 

ANSWERS. 

Problem. 

Gun. 

^          Cd^' 

d. 
In. 

If. 
Lbs. 

c. 

V. 

f.  s. 

K. 

log  K. 

colog  K. 

A         

3 
3 
4 
5 
5 
6 
6 

? 

7 
8 

10 
10 
12 
13 
13 
14 
14 

13 

13 

33 

50 

50 

105 

105 

105 

165 

165 

260 

510 

510 

870 

1130 

1130 

1400 

1400 

1.00 
l.no 
0.67 
1 .  00 
0.61 
0.61 
1 .  00 
0.61 
1.00 
0.61 
0.61 
1.00 
0.61 
0.61 
1.00 
0.74 
0.70 
0.70 

1150 
2700 
2900 
3150 
3150 
2600 
2S00 
2800 
2700 
2700 
2750 
2700 
2700 
2900 
2000 
2000 
2000 
2000 

1.4444 
1.4444 
3.0783 
2.0000 
3.2787 
4.7814 
2.9166 
4.7814 
3.3673 
5.5201 
6.6598 
5.1000 
8.3605 
9.9045 
6.6863 
9 . 0355 
10.2040 
10.2040 

0.1.5970 
0.15970 
0 . 48832 
0.30103 
0.51570 
0.67956 
0.46189 
0.67956 
0.52728 
0.74195 
0.82346 
0.70757 
0.92224 
0.99583 
0.82519 
0.95596 
1.00877 
1.00877 

9.84030—  10 

B   

9.84030—  10 

c 

9.51168  —  10 

D 

E 

F 

G 

H 

I 

9.69897  —  10 
9.48430  —  10 
9.32044—10 
9.53511—10 
9.32044  —  10 
9.47272—10 

J 

9.25805—  10 

K 

9.17654—10 

L 

9.29243—  10 

M 

9.07776—10 

N 

0.00417  —  10 

0 

9.17481  —  10 

P 

9.04404  —  10 

Q 

8.99123  —  10 

R 

8.99123—10 

, 

44 


EXTERIOR  BALLISTICS 


2.  Using  the  values  of  K  found  in  example  1  preceding,  determine  the  values 
of  8,  /,  and  log  C  for  the  conditions  given  in  the  following  table.  Correct  for  maxi- 
mum ordinate  or  for  height  of  target  according  to  rule,  but  consider  every  trajectory 
whose  time  of  flight  is  greater  than  five  seconds  as  requiring  correction  for  altitude. 


DATA. 

ANSWERS. 

Gun. 

Diff.  in 

Atmos- 

f3 

R'nge. 
Yds. 

Time  of 
flight. 

Max. 
ord. 

Value 
of 

ht.  of 
gun  and 

phere. 

5. 

f- 

lU 

log  (7. 

o 

d. 

If. 

T. 

Sees. 

Feet. 

log  Z. 

target. 

Bar. 

Ther. 

f^. 

In. 

Lbs. 

f.s. 

Feet. 

In. 

°F. 

A..  . 

3 

13 

1.00 

1150 

2600 

8.25 

277 

0.15970 

300 

.30.33 

24.7 

1.1011 

1.0050  0.12004 

B... 

3 

13 

1.00 

2700 

4400 

9.25 

366 

0.15970 

150 

30.13 

17.5 

1.1113 

1.0063  0.11660 

C,  . 

4 

33 

0.67 

2900 

3900 

5.10 

105 

0.48832 

200 

29.92 

15.7 

1.1080 

1.0037  0.44538 

T)  ,  . 

5 

50 

1.00  3150 

4300 

6.18 

154 

0.30103 

225 

29.83 

12.4 

1.1135 

1.0040  0.25606 

F„ 

F, 

50 

0.61  3150 

4300 

5.19 

108 

0.51570 

90 

29.57 

29.3 

1.0644 

1.0022  0.48956 

F..  , 

f) 

105 

0.61  2G00 

14800 

31.56 

4215 

0.67956 

1200 

29.45 

33.8 

1.0502 

1.0753  0.68982 

G 

6 

105 

1.00 

2800 

4000 

5.56 

124 

0.46489 

350 

29.37 

39.41 

1.0352 

1.0060,0.4.5247 

H    , 

6 

105 

0.61 

2800 

3700 

4.. 57 

85 

0.67956 

200 

29.07 

43.2 

1.0170 

1.0037  0.67384 

T 

7 

165 

1.00 

2700 

7000 

11.76 

563 

0.52728 

None 

28.95 

48.7 

1.0009 

1.0095  0.53100 

J 

7 

165 

0.61 

2700 

7400 

10.61 

455 

0.74195 

175 

28.83 

50.3 

0.9936 

1.00810.74824 

K 

8 

?m 

0.61 

2750 

8300 

11.49 

532 

0.82346 

450 

28.73 

52.8 

0.9852 

1. 0091  !o.  83387 

T„ 

10 

510 

1.00 

2700 

10100 

16.57 

1116 

0.70757 

500 

28.. 58 

69.3 

0.9475 

1.0193  0.73929 

IM 

10 

510 

0.61 

2700 

11000 

15.69 

997 

0.92224 

1100 

28.47 

95.7 

0.8936 

1.019010.97927 

N 

}?. 

870 

0.61 

2900 

23500 

37.61 

5758 

0.99583 

1500 

28.36 

97.4 

0.8867 

1.1072  1.09228 

0, 

13 

11.30 

1.00  2000 

10400 

21.53 

1889 

0.82519 

700 

28.27 

99.8 

0.8790 

1.0328  0.89522 

P    , 

13 

1130 

0.74,2000 

11300 

22.28 

2005 

0.95596 

508 

28.21 

74.8 

0.9243 

1.0351  1.00513 

Q. 

14 

1400 

0.70  2000 

14100 

28.36 

3264 

1.00877 

800 

28.20 

71.3 

0.9310 

1.058311.06443 

K... 

14 

1400 

0.70  2600 

14400 

21.83 

1925 

1.00877 

700 

28.71 

84.6  0.9225 

1.0335  1.05811 

3.  Given  the  measured  velocities  of  a  projectile  at  two  points,  as  determined  by 
experimental  firing,  as  given  in  the  following  table,  determine  the  resistance  of  the  air 
at  the  mean  velocity  between  the  two  points  of  measurement.  If  the  atmospheric 
conditions  at  the  time  of  firing  were  as  given,  what  would  be  the  corresponding 
resistance  under  standard  atmospheric  conditions? 


DATA. 

ANSWERS. 

DATA. 

ANSWERS. 

Problem. 

Projectile. 

Dist. 
between 
points  of 
measure- 
ment. 
Yds. 

Measured 
velocities  at. 

Rf. 
Lbs. 

Atmosphere. 

d. 
In. 

w. 
Lbs. 

Bar. 
In. 

Ther. 

°F. 

Its- 

First 
point. 

f.  s. 

Second 
point. 

f.s. 

Lbs. 

1   

3 

5 

6 

7 

12 

13 

14 

6 

13 

60 

105 

165 

870 

1130 

1400 

7b 

80 
90 
95 
100 
105 
110 
125 
200 

2650 
2250 
2550 
2680 
2870 
1910 
2540 
1951 

2600 
2200 
2500 
2580 
2800 
1880 
2460 
1874 

220.78 
767.76 
1444.5 
4492.2 
17022.0 
6045.5 
23188.0 
533.55 

28.00 
29.00 
30.00 
30.50 
31.00 
.30.00 
29.00 
29.53 

50 
60 
70 
80 
90 

i 

59 

228.55 

2 

783.43 

3 

1454.7 

4   

4539.8 

5 

17281.0 

6 

5257.0 

7..... 

22021.0 
533.55 

GENERAL  AND  APPROXIMATE  DEDUCTIONS 


45 


4.  Under  the  conditions  given  in  the  following  table,  compute  the  total  atmos- 
pheric, resistance  to  the  passage  of  the  projectile,  and  the  resultant  retardation  in 
foot-seconds. 


DATA. 

ANSWERS. 

Problem. 

Projectile. 

Velocity, 
f.  s.  ' 

Atmosphere. 

Retardation, 
f.s. 

Resistance. 
Lba. 

d. 
In. 

to. 
Lbs. 

c. 

Bar. 
In. 

Ther. 
°F. 

1 

3 

5 

6 

7 

12 

13 

14 

14 

3 

13 

60 

105 

165 

870 

1130 

1400 

1400 

13 

1.00 
1.00 
0.61 
0.61 
0.61 
0.95 
0.70 
0.70 
0.93 

2300 
1500 
1300 
2700 
2850 
1100 

850 
2500 

650 

30.00 
31.00 
29.00 
28.50 
29.45 
30.15 
28 .  67 
29.33 
30.40 

20 
55 
47 
82 
90 
95 
64 
75 
80 

493.39 

130.. 53 

44.23 

141.16 

86.92 

13.74 

3.43 

70.20 

12.54 

199  19 

2 

243.23 
144  "^4 

3 

4 

723.. 33 
2348.40 

482.27 
149  '^7 

5. 

6 

7 

8 

3059  90 

9 

5.06 

5.  What  is  the  resistance  of  the  air  to  a  baseball  of  3"  diameter,  weighing  8 
ounces,  moving  at  100  f.  s.;  supposing  the  resistance  of  a  sphere  to  be  1.25  times  that 
of  a  standard  ogival ;  and  what  would  be  its  retardation  ? 

Answers.     Resistance,      0.16337  pound. 

Retardation,  10.531  foot-seconds. 


6.  Given  the  data  in  the  following  tables,  compute  the  value  of  the  constant  A 
in  Mayevski's  formula,  for  each  individual  case. 


DATA. 

ANSWERS. 

Problem. 

Projectile. 

5. 

Velocity, 
f.  s. 

Resist- 
ance. 
Lbs. 

Retar- 
dation. 

f.  s. 

Value 
of 
a. 

Value 

d. 
In. 

tv. 
Lbs. 

c. 

of 
A. 

1 

6 

6 

6 

12 

14 

14 

70 

70 

70 

870 

1400 

1400 

1.00 
1.00 
1.00 
0.61 
0.70 
0.70 

1.0000 
1.0200 
1 . 0200 
0.9354 
0.9610 
0.9606 

1912.5 
1818.0 
1859.5 
2850.0 
850.0 
2500.0 

533.6 

491.8 

1248.6 

3052. 4 

86! 92' 
3.4328 

1.70 
1.70 
1.70 
1.55 
3.00 
1.70 

0.001259 

9 

0.00124023 

3 

0 . 0030296 

4 

0.0040649 

5 

0.000000059353 

6 

0.001248 

7.  A  6"  projectile,  weight  70  pounds,  is  fired  through  screens,  and  the  velocities 
measured  at  two  points  200  yards  apart  are  1951  f.  s.  and  1874  f.  s.  What  was  the 
mean  resistance  of  the  air?  Answer.     533.57  pounds. 


46 


EXTERIOE  BALLISTICS 


8.  A  6"  ogival-headed  projectile,  weight  70  pounds,  is  fired  through  screens  150 
yards  apart,  and  its  velocities  at  the  first  and  at  the  second  pairs  of  screens  are 
1846  f.  s.  and  1790  f.  s.,  respectively.  A  6"  flat-headed  projectile  of  the  same  weight 
is  fired  through  the  same  screens,  and  gives  velocities  of  1929  f.  s.  and  1790  f.  s., 
respectively.  What  was  the  resistance  of  each  projectile  ?  If  the  first  was  a  standard 
projectile,  what  was  the  coefficient  of  form  of  the  second  ? 

Ansivers.     First,  R=  491.82  pounds. 
Second,  E  =  1248.65  pounds. 
Coefficient  of  form  of  second  =  2.5388. 

9.  A  12"  projectile,  weight  850  pounds,  gave  measured  velocities  of  1979  f.  s. 
and  1956  f .  s.  at  points  500  feet  apart.  What  was  the  mean  resistance  of  the  air  ?  If 
the  density  of  the  air  at  the  time  of  firing  was  1.02  times  the  standard  density,  what 
would  be  the  resistance  in  a  standard  atmosphere  ? 

Ansivers.     Rf  =  2389.0  pounds.     J?«  =  2342.2  pounds. 

10.  Determine  the  resistance  of  the  air  to  and  the  consequent  retardation  of  a 
standard  3"  projectile,  weight  13  pounds,  when  moving:  (1)  at  2800  f.  s.;  (2)  at 
2000  f.  s. 

Answers.     (!)   Eesistance  =  250.34  pounds.     Eetardation=: 620.09  f.  s. 
(2)  Eesistance  =  142.67  pounds.     Eetardation  =  353.38  f .  s. 

11.  Determine  the  resistance  of  the  air  and  the  consequent  retardation  in  the 
following  cases.     (Standard  atmosphere ;  and  c=  1.00  in  each  case.) 


DATA. 

ANSWERS. 

Problem. 

Projectile. 

Velocity. 

f.  8. 

Resistance. 
Lbs. 

Retardation. 

d. 
In. 

w. 
Lbs. 

f.  s. 

1       

4 

4 

6 

G 

8 

8 

10 

10 

12 

12 

12.5 

12.5 

33 

33 
100 
100 
250 
250 
500 
500 
850 
850 
802.5 
1000.0 

2800 
2000 
2800 
2000 
2800 
2000 
2800 
2000 
2800 
2000 
1400 
1400 

445.1 
253.6 
1001.4 
570.7 
1780.2 
1014.5 
2781.5 
1585.2 
4005.4 
2282.7 
1251.6 
1251.6 

434.3 

2 

247.5 

3 

322.4 

4 

183.8 

229.3 

6 

130.7 

7 

179.1 

8 

102.1 

9    

151.7 

10 

86.5 

11 

50.2 

12 

40.3 

CHAPTEE  4. 

THE  EQUATION  TO  THE  TRAJECTORY  IN  AIR  WHEN  MAYEVSKI'S 
EXPONENT  IS  EQUAL  TO  2. 

New  Symbols  Introduced. 

p.  .  .  .Radius  of  curvature,  in  feet,  of  the  trajectory  at  any  point. 

A 

k.  . .  .The  ratio -^,  where  A  is  Mayevski's  constant,  and  C  is  the  ballistic 

coefficient. 
£.  . .  .The  base  of  the  Naperian  system  of  logarithms;  £  =  3.7183. 
n.  . . . The  ratio  of  the  range  in  vacuum  to  the  range  in  air  for  the  same 

angle  of  departure. 

67.  Mayevski's  equations,  as  given  in  (14)  and  (16),  show  that,  strictly  speak- 
ing, A  and  a  can  only  be  regarded  as  constants  in  the  general  expression  for  the 
retardation  caused  by  the  resistance  of  the  air 

dt  C  ^  ^ 

within  certain  limited  ranges  of  the  value  of  the  velocity,  v.  As  will  shortly  be  seen, 
in  the  attempt  to  derive  the  equation  to  the  trajectory  in  air,  it  will  be  possible  to 
succeed  in  cases  where  the  value  of  a  is  an  integer,  that  is,  for  cases  within  the  range 
of  the  formulse  in  (14)  where  the  value  of  v  lies  between  1800  f.  s.  and  1370  f.  s. 
(a  =  2),  or  within  the  lower  limits  where  a  =  3,  5  or  2  again;  and,  unfortunately, 
these  limits  do  not  include  the  initial  velocities  for  modern  high-powered  guns. 
This  is  true,  as  will  be  seen,  because  the  derivation  of  the  required  equation  involves 
the  integration  of  certain  expressions  in  which  the  value  of  a  must  appear  as  an 
exponent,  and  it  is  impossible  to  make  such  integrations  except  when  a  is  a  whole 
number.  The  equation  which  we  will  derive  will  therefore  only  be  correct  for  the 
limited  range  of  values  of  v  for  which  a  =  2.  Could  all  the  integrations  be  performed, 
for  the  decimal  as  well  as  for  the  integral  values  of  a,  a  series  of  equations  to  the 
trajectory  for  the  different  limits  of  the  initial  velocity,  could  be  derived  and  tabu- 
lated in  the  same  way  in  which  Mayevski's  expressions  for  retardation  shown  in  (14) 
were  tabulated,  but  as  it  is,  for  the  reason  given,  such  equations  cannot  be  derived  for 
the  initial  velocities  that  are  used  at  the  present  day;  namely,  from  1800  f.  s.  up, 
so  some  other  method  of  obtaining  solutions  must  be  found.  This  is  done  by  the  use 
of  certain  differential  equations,  as  will  be  explained  later. 

68.  Meanwhile  it  is  of  value  to  follow  through  the  derivation  of  the  equation  to 
the  trajectory  in  air  when  a  =  2,  both  for  educational  purposes  and  in  order  to  make  a 
comparison  with  the  equation  to  the  same  curve  in  vacuum.  We  must  remember, 
however,  that  this  equation  will  only  be  correct  for  initial  velocities  for  which  a=2 
in  (14),  and  that  for  all  other  initial  velocities  results  obtained  by  its  use  will  be  only 
approximate. 

69.  Assuming  that  the  axis  of  the  projectile  coincides  with  the  tangent  to  its  Forces 
path  at  every  point,  which  is  very  nearly  the  case  with  modern  rifled  guns,  the 
resultant  action  of  the  resistance  of  the  air  will  likewise  coincide  with  the  axis,  and 
the  trajectory  will  be  the  same  as  if  the  mass  of  the  projectile  were  concentrated  at  its 
center  of  gravity  and  moved  under  the  action  of  two  forces  only,  one  the  constant 


acting'. 


48 


EXTERIOR  BALLISTICS 


vertical  force  of  gravity,  w,  and  the  other  the  variable  resistance  of  the  air,  —  X  -yrv^, 

acting  in  the  tangent.  Figure  6  represents  the  trajectory,  which  is,  of  course,  a 
plane  curve,  under  the  foregoing  suppositions,  and  Figure  6(a)  represents  the  two 
forces  acting  upon  the  projectile  at  any  point,  the  resistance  of  the  air  being  denoted 

by  —  /,  in  which  /  is  the  retardation,  -^,  which   we  are  now  taking  as  proportional 

to  v^ 


dt 


Figure  6 


70.  Taking  vertical  and  horizontal  axes  at  the  point  of  departure,  0,  let  V  be 

the  initial  velocity,  cf)  the  angle  of  departure,  v  the  velocity  at  any  point  whose 

coordinates  are  (x,  y),  and  Vn  the  horizontal  component  of  the  velocity  at  that  point. 

A 
p  is  the  radius  of  curvature  of  the  curve  at  that  point.    Then,  letting  lc=  -T^in  equa- 

dv 
tion  (18),  we  can  put -,-7- =  —  A;i'-,  and,  since  w  has  no  horizontal  component,  the 

acceleration  parallel  to  the  axis  of  X  is  given  by 

d-x 


dr- 


=  — kv-  cos  6 


but 


d^x 


df       dt 


*    V  cos  d  =  Vh,  and  v=  -^j  ;  whence  (19)  may  be  written 


dt 


dvj, 
dt 


—  kvi, 


ds 
IF' 


dvn  _ 


=  —  kds 


(19) 

(20) 
(21) 


and  integrating  (21)  between  corresponding  limits  of  Vh  and  s  we  get 

I-'cos  0 

t)ft=7cos<^€-^'»        (22) 


lOffef 


ge  ^h 


=  —ks 


n 


Fcos  d> 

loge   =ks 


Next  resolving  along  the  normal,  since  the  acceleration  towards  the  center  of  curva- 

ture  IS  given  by  the  expression  — ,  p  being  the  radius  of  curvature  at  that  point, 

P 
we  have 


—  g  cos  6 


(23) 


GENERAL  AND  APPEOXIMATE  DEDUCTIONS        4.9 


ds 
But  v  =  Vh  sec  9,  and  p=  — j^,  whence  (23)  may  be  written 

do 


Vh^  sec-  0 cW=—g  cos  6  ds=  —gd.v  (as  dx  =  ds cos  6) ;  sec-  6  d6=  —  ^ -^        (24) 


gdx 


Now  substituting  in  (2-i)  the  value  of  vi,  given  in  (22),  we  get 

sec-  ede=-  Yr^^—^  €-'''dx  (25) 

V^  cos^  </> 

71.  In  the  case  of  the  flat  trajectory,  in  which  the  angle  of  departure  does  not 
exceed  -t°  or  5°,  the  difference  between  the  values  of  s  and  x  is  so  small  that  it  may  be 
practically  disregarded,  and  x  may  be  substituted  for  s  in  (25),  giving,  after 
integration  * 


tan^ 


tan  ^  =  tan  0-   — ^/— ^  (e^*^-l)     (26) 


0  2kV-  cos^  <l> 

But  c-^-^j  when  expanded  by  j\laclaurin's  theorem,  equals  f 

1  +  21CX  +  2k-x-  +  ^k^x^  + 

so  that  e"^-^  - 1  =  2 A-.r  ( 1  +  kx  +  § fc^T-  + ) 

whence,  substituting  in  (26)  and  writing  ^  for  tan  6,  we  have 

or,  integrating  between  corresponding  limits  of  x  and  y, 

y  =  xtancf>-  gy^^^y^  (l  +  t^-'^  +  ^^-'-^'+ •  •  •  •)  (^7) 

But  the  greatest  value  of  kx  is  always  a  small  fraction  in  any  trajectory  flat  enough 
to  justify  the  substitution  of  x  for  s  which  has  already  been  made;  hence  we  may 
neglect  the  terms  beyond  k'x-  in  the  expansion,  and  write  for  the  equation  to  the 
trajectory  in  air  when  a  =  2  Equation  to 

trajectory  in 


r.2 


^  =  .r  tan  c^-  ^f^  (l  +  ^kx  +  ik^x^)  (28)    ^t'"' 

*  The  integration  in  paragrapli  71  is  as  follows:     From  (25) 


From  calculus  we  know  that  ( sec-  d  dO  =  tan  6  +  d,  C^  being  the  constant  of  integration. 
From  calculus  we  know  that  fe"  cf?/  =  e*  +  C'j,  C,  being  the  constant  of  integration.  Now 
let  y  =  2kx  and  the  above  becomes 

whence  f  1    f  „,   ,  .„,  1      ,        „ 

The  integration  between  the  limits  given  above  therefore  becomes 

(tan  .  +  C.)  -  (tan  ^  +  C.)=-  -,.^L   p^  ,.-  +  c,)  -  (^  .'  +  C.)] 

tan  ^  z.  tan  <^  -  y^^^^  X  ^  (e=*-  l) 
t  The  expansion  in  paragraph  71  is:     From  either  algebra  or  calculus  we  have  that 
«^  =  1  +  2/ +  if- +  ll^  +  ||- +•  •  • -etc. 

whence,  if  we  let  y  ^  2kx  we  have 

Ak-x'   ,     %k^x^    ,       16k*x' 
e^fc-  =  1  -f-  2kx  +  -2~  +  3^^  +  4  X  3  X  2  +  •  •  •  ■^^''- 

or  e-^x  =  1  ^  2A-J-  (1  -\- kx  +  ^  k^'x'  +  |A:  V  + etc.) 


50 


EXTERIOR  BALLISTICS 


The  ratio  n 


72.  Comparing  this  equation  with  (2),  it  will  be  seen  that  its  first  two  terms 
represent  the  trajectory  in  vacuum,  and  that  it  only  differs  from  the  latter  by  having 
other  terms,  subtractive  like  the  second,  and  containing  higher  powers  of  k  and  x. 

73.  The  value  of  k  in  the  equation  to  the  trajectory  in  air  (28)  just  deduced  is, 

as  already  stated,  — ^ 


Assumptions 
made. 


c= 


Scd- 


where  A  is  the  experimentally  determined  coefficient,  and 
is  the  ballistic  coefficient.     As  a  matter  of  general  interest,  it  may  be 


stated  that,  for  the  value  of  .4  =  0.0001316,  the  value  assigned  to  A  by  Mayevski  when 
a  =  2,  the  value  of  k  for  our  naval  guns  from  the  6-pounder  up  to  the  13"  gun,  varies 
from  about  0.00011006  to  about  0.00002022,  for  the  standard  projectile  and  standard 
density  of  the  air. 

74.  If  we  put  y  =  0  in  equation  (28),  we  get  for  values  of  x^  one  equal  to  zero, 
denoting  the  origin,  and  another,  the  range  X,  given  by 

V-  sin  2(^ 


Z(l  +  pX  +  iFZ^)  = 


9 


(29) 


But  the  second  member  of  (29)  is  the  range  in  vacuum  for  the  same  initial  velocity, 
V,  and  the  same  angle  of  departure,  (f>,  so  we  see  that  the  expression 

l  +  ikX  +  WX^=  Z(invacuum) 
■^  -^  Z(mair) 

This  ratio  will  be  found  to  play  an  important  part  in  many  ballistic  problems,  and 
will  hereafter  be  designated  by  the  letter  n.  Hence  we  have  for  the  range  in  air  the 
expression 

-^^71sin2^  (30) 

gn 


(31) 


or,  if  it  be  desired  to  find  the  value  of  n  for  a  given  range, 

_  72  sin  2<f> 

75.  Since  k  is  a  very  small  fraction,  the  value  of  ti,  which  is  evidently  unity  for 
X  =  0.  increases  slowly  with  X,  and  for  moderate  values  of  X  is  only  slightly  greater 
than  unity.    These  deductions  follow  from  the  form  of  the  equation 

l  +  ^kX  +  ik'X^  =  n 

76.  It  is  Avell  to  summarize  here  that  the  following  suppositions  have  been  made 
in  deriving  the  equation  to  the  trajectory  in  air  when  a  =  2,  and  these  suppositions 
must  be  held  to  be  correct  in  all  consideration  of  this  equation ;  and  the  equation  is 
inaccurate  to  whatever  degree  results  from  the  lack  of  correctness  of  any  one  or 
more  of  these  assumptions : 

1.  That  a=2,  and  that  the  corresponding  value  of  A  is  correct. 

2.  That  the  axis  of  the  projectile  coincides  with  the  tangent  to  the  trajectory  at 
every  point,  and  that  the  resistance  of  the  air  will  therefore  act  along  the  same  tangent. 

3.  That  the  curve  is  so  flat  that  we  may  consider  dx=ds  without  material  error. 
•4.  That  kx  is  so  small  in  value  that  any  term  involving  powers  higher  than  k'x^ 

may  be  neglected. 

77.  The  following  examples  show  the  form  for  work  under  the  formulae  derived 
in  this  chapter.  It  must  be  remembered,  be  it  again  said,  that  these  formulas  are 
derived  on  the  assumptions  given  in  the  preceding  paragraph,  and  results  obtained 
by  their  use  are  therefore  only  approximately  correct  for  the  usual  present-day  initial 
velocities. 


GENERAL  AND  APPROXIMATE  DEDUCTIONS         51 

For  a  6"  gun,  given  that  the  initial  velocity  is  2600  f.  s.,  and  that  an  angle  of 
departure  of  4°  14'  30"  gives  a  range  of  7000  yards,  to  compute  the  value  of  the  ratio 
between  the  ranges  in  vacuum  and  in  air  for  that  angle  of  departure;  that  is,  the 
value  of  n. 

_  V-  sin  2^ 

7  =  2600   log  3.41497 2  log  6.82994 

2(^  =  8°  29'  00"    sin  9.16886-10 

g  =  32.2    log  1.50786 colog  8.49214-10 

Z  =  21000    log  4.32222 colog  5.67778-10 

n  =  1.4748    log  0.16872 

78.  Given  that  the  angle  of  departure  for  the  12"  gun  of  2900  f.  s.  initial 
velocity  (^  =  870  pounds;  c  =  0.61)  for  a  range  of  10,000  yards  is  4°  13'  12",  compute 
the  approximate  value  of  the  ordinate  at  a  distance  of  2000  yards  from  the  gun,  and 
compare  it  with  the  ordinate  in  vacuum  at  the  same  point  for  the  same  angle  of 
departure. 

In  vacuum  y  =  x  tan  S  —  7:^7^^ — ^— 

•^  2V^  cos^  (^ 

T      •  ,       .  nqx^         1  V^  sin  2d> 

In  air  y  =  x  ia.xi.  dt  —  ~^^ — 5— where  n=: =^— £- 

2  y^  cos^  ^  gX 

TForfc  in  Vacuum. 

a;=6000    log  3.77815 2  log  7.55630 

<^  =  4°  13'  12"    tan  8.86797-10.  .      sec  0.00118 2  sec  0.00236 

^  =  32.2    log  1.50786 

2    log  0.30103 colog  9.69897-10 

7  =  2900    ..log  3.46240 2  log  6.92480.. 2  colog  3.07520-10 


442.710    lo<r  2.64612 


69.294    log  1.84069 


!/  =  373.416 

Vi'ork  in  Air. 

From  work  in  vacuum;  a;  tan  <i  =  442.71  and  ^ ^^^ ^^  ,      =69.294. 

'  ^  272cos2<^ 

7  =  2900    log  3.46240 2  log  6.92480 

2<^  =  8°26'24"    sin  9.16665-10 

g  =  32.2    colog  8.49214-10 

Z  =  30000    log  4.47712 colog  5.52288-10 


jx' 


2V^cos^cf, 
ngx^ 
'^  cos^  <^ 

a;  tan  (^  =  442.710 


:1.2778    log  0.10647 

:69.294    lo£  1.84069 


»T^f—.     =   88.544    log  1.94716 


?/  =  354.166 

n   r     ^     ^onnn        i  fin  vacuum 373.416  feet. 

Ordniate  at  2000  yards \  ^       .  .- ,  ■^r.n  j-    j. 

''  In  air 3d4.166  feet. 


52 


EXTERIOR  BALLISTICS 


EXAMPLES. 

1.  Determine  the  value  of  the  radius  of  curvature  of  the  trajectory  at  the  point 
of  departure  for  a  muzzle  velocity  of  3000  f.  s.,  and  an  angle  of  departure  (1)  of 
3°,  and  (2)  of  8°. 

Atiswers.     For  3°,  124,390  feet.    For  8°,  125,443  feet. 

2.  In  the  two  cases  given  in  Example  1  preceding,  the  striking  velocities  are 
1600  f.  s.  and  1240  f.  s.,  respectively,  and  the  angles  of  fall  are  3°  29'  and  11°  08', 
respectively.    Compute  the  radii  of  curvature  at  the  point  of  fall. 

Answers.     For  3°,  79,648.5  feet.    For  8°,  48,667.0  feet. 

3.  Using  the  equation  to  the  trajectory  in  air  when  a  =  2,  compute  the  value 
of  n  and  the  approximate  angle  of  departure  in  each  of  the  following  cases : 


DATA. 

ANSWERS. 

Problem. 

Gun. 
In. 

Value  of  k. 

Initial 
Velocity. 

f.  s.  ■" 

Range. 

Yds. 

n. 

Angle  of 
departure. 

1 

6 

6 
6 

8 
8 
8 

0.00004738 
0.00004738 
0.00004738 
0.00003369 
0.00003369 
0.00003369 
0.00002632 
0.00002632 
0.00002632 
0.00002229 
0.00002229 
0.00002229 

2400 
2400 
2400 
2400 
2400 
2400 
2400 
2400 
2400 
2400 
2400 
2400 

1000 
2000 
3000 
1000 
2000 
3000 
1000 
2000 
3000 
1000 
2000 
3000 

1.101 
1.216 
1 .  345 
1.071 
1.148 
1.233 
1.0.55 
1.114 
1.177 
1.046 
1.095 
1.147 

0°     32' 

2 

1       10 

1       56 

4 

0       31 

5 

1       06 

6 

1       47 

7 

10 
10 

0      30 

8 

1       04 

9 

10 

11 

12 

10 
12 
12 
12 

1       42 

0  30 

1  30 
1       39 

4.  A  6"  gun  with  2900  f.  s.  initial  velocity  gave  a  measured  range  of  5394  yards 
for  an  angle  of  departure  of  3°  03'  51".    Compute  the  value  of  n  from  the  firing. 

Answer.     n=  1.72290. 


5.  A  6"  gun  with  2900  f.  s.  initial  velocity  gave  a  measured  range  of  2625  yards 
for  an  angle  of  departure  of  1°  07'  49".    Compute  the  value  of  n  from  the  firing. 

Answer.     n  =  1.308. 


GENERAL  AND  APPROXIMATE  DEDUCTIONS 


53 


6.  Given  the  data  in  the  first  six  columns  of  the  following  table,  compute  the 
ordinates  of  the  trajectory  for  each  of  the  given  abscissee,  in  both  vacuum  and  air, 
using  the  equation  to  the  trajectory  when  a  =  2.  In  working  in  air,  first  determine 
the  value  of  n  for  the  given  range  corresponding  to  the  given  angle  of  departure,  then 
determine  the  value  of  k  from  this  by  using  the  formula  n  =  l  +  5fcX  (neglecting  the 
square  and  higher  powers  of  hX),  and  the  required  ordinates  by  the  use  of  the  value 
of  k  thus  found : 


DATA. 

ANSWERS. 

Problem. 

Gun. 
In. 

Initial 

velocity. 

f.  s. 

Weight  of 
pro- 
jectile. 
Lbs. 

Angle 
of   depar- 
ture. 

Range. 
Yds. 

Abscissa. 
Yds. 

Ordinates  in  feet. 

Vacuum. 

Air. 

1. 

12 

12 

12 

12 

12 

() 

G 

G 

3 

3 

3 

3 

2800 
2800 
2800 
2800 
2800 
2000 
2900 
2900 
2800 
2800 
2800 
2800 

850 

850 

850 

850 

850 

100 

100 

100 

13 

13 

13 

13 

2°     11' 
2       11 
2       11 
2       11 
2       11 
1       17 
1       17 
1       17 
1       01 
1       01 
1       01 
1       01 

5000 
5000 
5000 
5000 
5000 
3000 
3000 
3000 
2000 
2000 
2000 
2000 

1000 
2000 
3000 
4000 
5000 
1000 
2000 
3000 
500 
1000 
1500 
2000 

95.9 

154.7 

176.5 

161.4 

109.2 

50.0 

65.5 

46.5 

22.0 

34.8 

38.3 

32.5 

95.0 

2 

147.7 

3 

1.53 . 0 

4 

105.5 

5 

000.0 

6 

48.2 

7. 

51.7 

8.    

00.0 

9 

21.5 

10 

.30.7 

11 

24.5 

12 

00.0 

CHAPTER  5. 

APPROXIMATE  DETERMINATION  OF  THE  VALUES  OF  THE  ELEMENTS  OF 
THE  TRAJECTORY  IN  AIR  WHEN  MAYEVSKI'S  EXPONENT  IS  EQUAL 
TO  2.  THE  DANGER  SPACE  AND  THE  COMPUTATION  OF  THE  DATA 
CONTAINED  IN  COLUMN  7  OF  THE  RANGE  TABLES. 

New  Symbols  Introduced. 
h.  . . .  Height  of  target  in  feet. 

S .  . . .  Danger  space  in  feet  or  yards  according  to  work. 
79.  Before  presenting  for  discussion  the  more  exact  computations  of  the  ele- 
ments of  the  trajectory  by  the  use  of  the  ballistic  tables,  we  will  in  this  chapter 
deduce  formulae  by  means  of  which  the  values  of  those  elements  can  be  determined 
with  a  sufficient  degree  of  approximation  for  certain  purposes;  and  it  will  be  seen 
later  that  a  few  of  these  formulae  are  sufficiently  exact  in  their  results  to  enable  us  to 
use  them  practically.  Eemember  that  the  inaccuracies  in  these  formulae  result  from 
the  assumptions  upon  which  the  derivation  of  the  equation  to  the  trajectory  in  air 
was  based,  as  enumerated  in  paragraph  76  of  the  preceding  chapter.  For  our  present 
purposes  we  will  take  as  the  equation  to  the  trajectorj'  in  air  the  one  given  in  (28), 
but  simplified  by  the  omission  of  higher  powers  of  lex  than  the  first.  The  equation 
then  becomes 

y  =  xtan^-^^^{l  +  ilcx)  (33) 

In  this  equation  Jc  can  no  longer  be  considered  as  strictly  constant,  but  its  value, 
when  found  for  any  one  value  of  <f>,  may  be  used  over  a  considerable  range  of  values 
of  (f>,  since  it  increases  slowly  with  mcreases  of  range,  provided  the  trajectory  be 
reasonably  fiat.  We  shall  still  denote  by  n  the  ratio  of  the  range  in  vacuum  to  the 
range  in  air,  which  is  now  given  by 

Approximate  80.  To  determine  the  approximate  horizontal  range,  put  y  =  0  in    (32),  and 

"'^'range.    solve  for  X.    An  X  factor  will  divide  out,  so  one  value  of  x  is  zero,  for  the  origin,  as 
was  to  be  expected,  and  the  remaining  equation  is 

It  is  at  once  apparent  that  this  is  an  awkward  equation  for  logarithmic  work,  and 
furthermore  not  very  accurate  for  work  with  five  place  logarithmic  tables  owing  to  the 
decimal  value  of  Jc. 
Angle  of  fall.  81.  Differentiating  (32)  we  get 

4^  =  tan  ^  =  tan  <^  -  -^^^-^  ( 1  +  hx)  (34) 

ax  V-  cos-  ^ 

But  the  angle  of  fall,  w,  is  the  negative  of  the  value  of  0  at  the  point  of  fall,  where 
x  =  X,  hence 

tan  w  =  —  tan  d>  +  ^^^^ — .^ —  (1  +  TcX) 

V-  COS"  <^ 

,    .        2 y ^  sin  (f>  COS  ^        V   ,1  •  1  ■.. 

and  since -=7iX,  this  may  be  written 

9 

.  4-       ,  ,   2tan<i/-,   ,  7  T-x      4.       ,  f2  +  21cX  —  n\ 

tan  0)  =  —  tan  4>  H ^  ( 1  +  A:A )  =  tan  <b   — ■ 

n  \  n         J 

Also  from  l  +  §/tA'=n  we  get  2A;Z  =  3(w  — 1),  whence 

tanw  =  tan</>[2-  -j  (35) 


GENEEAL  AND  APPROXIMATE  DEDUCTIONS        55 

From  (35)  we  see  that  the  angle  of  fall  is  always  greater  than  the  angle  of  departure, 

but  can  never  reach  double  the  latter.* 

dr 
82.  Eeturning  to  equation  (32)  and  writing—,'-  for  Vh,  and  x  for  s,  we  have         Time  of 


=  T'  cos  </)e-'^^ 


dt 
Separating  the  variables  and  integrating  between  corresponding  limits  f 

{^  (^■■'■dx  =  V  cos  <^  ^dt  ^^'!^~  -^  =  ry  cos  </) 

Jo  Jo  "^ 

Expanding  e'''-^  and  neglecting  higher  powers  than  the  second  X 

xll+^]  =  TVcoscf> 

but  from  l  +  -p.Y  =  n  we  get  that  1  +  ^  = -^^^ ,  therefore 

y^3n_fi^        Z^  (36) 

4  K  cos  ^ 

*  This  follows  from  the  form  of  the  expression,  for  from  paragraph  75  we  know  that 
7^  =  1  +  gfcX  +  Jfc-X'',  from  which  we  see  that  n  is  unity  when  .Y  =  0  and  increases  very 

slowly  with  X,  k  being  a  very  small  decimal.    Therefore        is  always  less  than  unity  and 

2 is  always  greater  than  unity;  and  the  angle  of  fall  must  therefore  always  be  greater 

than  the  angle  of  departure.     Also  as  n  must  always  be  greater  than  unity  for  any  real 
range,  then  —  must  always  be  a  positive  real  number,  and  therefore  the  value  of  2 

must  always  be  less  than  2;  therefore  the  angle  of  fall  can  never  become  twice  as  great  as 
the  angle  of  departure. 

f  The  integration  in  paragraph  82  is  as  follows:    From  integral  calculus 


Uvdu^eV  +  C 

C  being  the  constant  of  integration.    Now  let  y  =  A:.T  and  we  have 
L^^dX  =  -^   t^^d  (kX)  =  -j-  (t^^)  +  C 

Therefore  e*^di=Fcos0      dt 

Jo  Jo 

becomes  (^+  o]  —  U^  +  A  =  (V  cos  4>  X  T  +  C,)  —  (V  cos  4>  X  0  +  C,) 

Ci  being  the  constant  of  integration  in  the  second  term.    The  above  becomes 

— ^ —  —  YT  cos  (p 

t  The  expansion  of  e*-r  following  the  integration  is  as  follows:     From  either  calculus 
or  algebra  we  know  that 

.vz=l  +  7/  +  -||-  +  -^  +  ....etc. 

and  substituting  kX  for  y,  and  neglecting  the  higher  powers  than  the  square,  we  have 

e''^=l  +  kX  +  ^ 
whence  ^^j  _  1  =  fcx  +  -^ 


whence  X  (l  +  ~]  =  VT  cos  cp 


66  EXTEEIOE  BALLISTICS 

nX 


As  the  range  in  vacuum  for  the  same  values  of  V  and  (^  would  be  nX,  then 


Fees  ^ 

is  the  time  of  flight  in  vacuum,  and  so  we  see  that  the  time  of  flight  in  air  is  always 
less  than  it  would  be  in  vacuum,  approaching  three-fourths  the  latter's  value  as  a 
minimum.*     For  flat  trajectories,  cos  </>  may,  of  course,  be  taken  as  unity. 
Remaining  83.  Equation  (32),  with  the  substitution  of  x  for  s,  gives  the  value  of  the  hori- 

velocity.  i  \       /  7  .  . 

zontal  component  of  the  velocity  at  any  point  m  the  trajectory,  and  smce  the  strikmg 
velocity  is  the  horizontal  velocity  at  the  point  of  fall  multiplied  by  sec  w,  we  have 

V  cos  d> 
'^  f.'^-^  cos  w 

whence,  putting  for  t^^  the  first  two  terms  of  its  expansion,  and  calling  ?  equal 

to  unity. 

The   striking   velocity,   therefore,   is   always   less   than   the   initial   velocity,   being 
reduced  to -^  when  n  =  3,  a  value  which  it  seldom  reaches.f 
Coordinates  84.  Since  the  trajectory  is  horizontal  at  its  highest  point,  we  obtain  the  abscissa 

of  vertex.  . 

{xq)  of  the  vertex  by  putting  6  =  0  in  (34),  thus  getting ?  =XQ{l-\-'kxQ), 

and  since "^ =nX,  this  may  be  written  Xq{1-\-1cXq)  =  -^r-,  a  quadratic  equa- 

9  '^ 

tion,  the  solution  of  which  gives  iCo  =    ~  97/" — ^    which,  since  A;Z  =  |(n  —  l), 

may  be  written 

if  we  divide  both  numerator  and  denominator  of  the  second  term  of  (38)  by  w  we  get 


_  Vl  +  3n(n-l)-l  ^  .33. 

'''-  3(/i-l) ^  .          ^'^^^ 


/ 


1    +3-A-J- 
_       n"  n         n 

3- A 

n 


from  which  we  see  that  when  n  is  infinity,  the  value  of  x^  becomes— ^  =0.58-3r, 

*  We  know  that  n  =  1  +  ifcX  +  Jfc^'Z^  therefore  n  increases  slowly  with  the  range  and 
is  always  greater  than  unity.    Therefore  3n  is  always  greater  than  3  and  3»  +  1  is  always 

greater  than  4  but  less  than  4n.    Therefore  —  7^-     is  always  less  than  unity,  and  as 


4w  Fcos0 

is  the  time  of  flight  in  vacuum,  the  time  of  flight  in  air,  which  is  T  =      ."'"     X  y- —     must 
be  always  less  than  ^ r  ;  that  is,  the  time  of  flight  in  air  is  always  less  than  it  would  be 

KCOS^ 

in  vacuum.    We  also  see  that   ^^        ^T^  +  ;i^  =^r  +  T- ,  and  therefore  that  if  n  becomes 

in  4n.      4m        4       4» 

infinitely  great  then      p"—  becomes  -j-,  which  is  evidently  the  maximum  possible  value 

of     ^7^      ;  and  we  therefore  see  than  the  time  of  flight  in  air  cannot  be  less  than  three- 
4?t 

fourths  of  the  time  of  flight  in  vacuum. 

f  As  in  the  note  to  paragraph  82,  n  is  necessarily  greater  than  unity,  therefore  3n  —  1 

2 
Is  necessarily  greater  than  2,  and  „    ■,   is  therefore  necessarily  less  than  unity;  there- 
fore the  final  velocity  is  always  less  than  the  initial  velocity. 


GENEEAL  AND  APPEOXIMATE  DEDUCTIONS        57 

from  which  we  see  that  the  abscissa  of  the  vertex  is  always  greater  than  half  the  range 
but  never  reaches  0,58X,* 

The  ordinate  of  the  vertex  may  be  obtained  by  substituting  the  value  of  Xq 
obtained  from  (38)  in  the  equation  to  the  trajectory  (32),  but  an  equally  accurate 
and  much  sim^Dler  determination  is  given  by 

!/.=  2|-"  (39) 

This  assumes  the  height  of  the  vertex  to  be  that  from  which  a  body  would  fall  freely 
(in  vacuum)  in  half  the  time  of  flight.  Actually  the  vertical  velocity  of  the  pro- 
jectile is  reduced  by  air  resistance,  but  since  the  time  the  projectile  takes  to  describe 
the  descending  branch  of  the  trajectory  is  somewhat  greater  than  half  the  time  of 
flight,  the  air  resistance  is  approximately  allowed  for  by  equation  (39). 

85.  As  an  example  of  the  use  of  the  foregoing  formula,  we  will  compute  the 
various  elements  of  the  500-yard  trajectory  of  the  old  pattern  United  States  magazine 
rifle,  for  which  the  initial  velocity  is  2000  f.  s.,  and  the  angle  of  departure  for  the 
given  range  is.  0°  31'  35".  After  finding  the  value  of  n,  we  may,  as  the  angles  are 
small,  use  ^  and  w  in  place  of  their  tangents,  and  the  formulae  then  are 

y2sin2(^  [^       1\^     rp      3n  +  l  ^  Z   .  ^,  _      2       ^. 

_  Vl  +  3/K/^-l)-l  Y-  ,/  -  91^ 
3(n-l)  ^^'  ^"~    8 

7  =  2000    log  3.30103 2  log  6.60206 

2</>  =  l°  03'  10"    sin  8.26418-10 

(/  =  32.2    log  1.50786 colog  8.49314-10 

Z  =  1500    log  3.17609 colog  6.82391-10 

n  =  1.5216    log  0.18229 

—  =  0.6572    colog  9.81771-10 

n 

2 — ^  =  1.3428  log  0.12801 

3?;  =  4.5648 

3?z.-M  =  5.5648    log  0.74545 

3^-1  =  3.5648    ..log  0.55204 colog  9.44796-10 

<^  =  31.6'    log  1.49969.      sec  0.00002 

4   log  0.60206 colog  9.39794-10 

Z=1500    log  3.17609 

7  =  2000    log  3.30103 colog  6.69897-10.     log  3.30103 

2    log  0.30103 


:42.43'    lo2  1.62770 


r=  1.0434    lo£T  0.01847 


i'a,  =  1122    loff  3.05002 


*  Let  us  suppose  that g  .    ^. ="9' •    Solving  this  quadratic  for  n,  we  find 

that  under  these  conditions  »  =  1;  and  therefore  if  n  be  greater  than  unity  the  value  of  the 
left-hand  member  above  must  be  greater  than  I;  therefore  the  value  of  Xo  must  always  be 

greater  than  -^  for  a  trajectory  in  air;  and,  as  x^  =  O.SSi  when  n  is  infinity,  we  also  see 
that  the  value  of  x^  can  never  reach  0.58Z. 


58  EXTEEIOR  BALLISTICS 

n  =  1.5216 

3n  =  4.5648  ..log  0.65942 

w-l  =  0.5316  ..log  9.71734-10 

3(n-l)  =1.5648 log  0.19446 colog   9.80554-10 

3n(w-l)  =2.3810  ..log  0.37676 

l  +  3n(n-l)  =3.3810  ..log  0.52905 

Vl  +  3n(n-l)  =1.8388.^  log  0.26453 


Vl  +  3n(n-l) -1  =  0.8388   log  9.92366-10 

X  =  1500    log  3.17609 

a:o  =  804.05    log  2.90529 

T  =  1.0434   .  .log  0.01847 2  log  0.03694 

^  =  32.2    log  1.50786 

8    \os  0.90309 coloff  9.09691-10 


^0=^4.3824    log  0.64171 

a)  =  0°  42'  24".  a-o  =  368.02  yards. 

T=  1.0434  seconds.  ?/o  =  4.3824  feet. 

i?,,=  1122f.  s. 

86.  As  another  example,  take  the  ease  of  the  6"  gun  with  7"  =  2400  f.  s.,  and  an 
angle  of  departure  of  2°,  for  which  the  range  is  3100  yards. 

n=       g^        ;  tana>  =  tan<^(^2--j,  T- -^  ^  V^o^  '  ^-"Sr^^^ 

y  =  2400    log  3.38021 2  log  6.76042 

2</>  =  4°    sin  8.84358-10 

^  =  32.2    log  1.50786 colog  8.49214-10 

Z  =  9300    log  3.96848 colog  6.03152-10 

ri  =  1.3417    log   0.12766 

—  =  0.7453    colog   9.87234  —  10 

n 

2-  -=1.2547    log  0.09854 

n 

3n  =  4.0251 

3n  +  l  =  5.0251    log  0.70115 

3/1-1  =  3.0251    ..log  0.48074 colog   9.51926-10 

cj)  =  2° tan  8.54308-10.  .      sec  0.00026 

4 log  0.60206 colog  9.39794-10 

Z  =  9300 log  3.96848 

7  =  2400   log  3.38021 colog  6.62979-10.  .     log   3.38021 

2    ." log   0.30103 

a,  =  2°  30'  38"    tan  8.64162-10 


r  =  4.871    log  0.68762 


t;,  =  1586.7    log   3.20050 

w  =  2°   30'  38". 
r= 4.871  seconds. 
r„=  1586.7  f.  s. 


GENERAL  AND  APPROXIMATE  DEDUCTIONS 


59 


87.  If  the  value  of  n  be  known,  A;  may  be  found  from  n  =  1  +  '^kX,  and  then  by 
substituting  the  proper  value  of  x  in  the  equation  to  the  trajectory  (32),  the  corre- 
sponding value  of  y,  the  ordinate  of  the  trajectory  at  a  distance  x  from  the  gun  may 
be  computed ;  and  this  was  the  way  in  which  the  last  problems  in  the  examples  under 
the  last  chapter  were  worked.  If,  however,  we  know  the  angles  of  departure  corre- 
sponding to  various  ranges  (which  data  is  contained  in  the  range  table  for  the  gun) 
the  approximate  value  of  y  for  any  value  of  x,  for  the  trajectory  for  a  given  range, 
may  be  more  readily  found  as  follows.  Referring  to  Figure  7,  let  (x',  y')  be  the 
coordinates  of  the  point  M  on  the  trajectory  for  which  <j>  =  \p'  +  p.  Then,  by  the 
principle  of  the  rigidity  of  the  trajectory,  if  the  angle  of  departure  were  i/'',  the 
horizontal  range  would  equal  OM,  or  what  is  practically  the  same  thing,  x'.  Conse- 
quently, if  we  taken  from  the  range  table  the  angle  of  departure  for  a  range  x',  and 
subtract  it  from  the  angle  of  departure  for  the  given  trajectory,  the  result  will  be  the 
angle  /;.    Then  y  —x  tan  [). 


Figure  7. 


88.  By  the  term  "  danger  space  "  is  meant  an  interval  of  space,  between  the 
point  of  fall  and  the  gun,  such  that  the  target  will  be  hit  if  situated  at  any  point  in 
that  space.  In  other  words,  it  is  the  distance  from  the  point  of  fall  through  which  a 
target  of  the  given  height  can  be  moved  directly  towards  the  gun  and  still  have 
the  projectile  pass  through  the  target.  Therefore,  within  the  range  for  which  the 
maximum  ordinate  of  the  trajectory  does  not  exceed  the  height  of  the  target,  the 
danger  space  is  equal  to  the  range,  and  such  range  is  known  as  the  "  danger  range.'' 
Referring  to  Figure  8,  AH  =  S  is  the  danger  space  for  a  target  of  height  AB  —  h,  in 
the  case  of  the  trajectory  OBH.  It  will  be  seen  from  Figure  8  that,  when  the  value 
of  li  is  very  small  in  comparison  with  the  range,  the  danger  space  is  given  with 
sufficient  accuracy  by  the  formula 

*S  =  /icotw  (40) 


Danger 
space. 


60 


EXTERIOR  BALLISTICS 


This  assumes  that  the  tangent  at  the  point  of  fall  is  identical  with  the  curve  from 
H  to  B,  whereas  it  really  passes  above  B,  so  that  the  result  given  by  (10)  is  somewhat 
too  small. 


—  X. 


Figure  8. 

89.  A  more  accurate  formula  for  the  danger  space  is  deduced  as  follows :  Calling 
the  very  small  angle  AOB^^fj),  we  have,  very  nearly,  tan  A<^=  ^  and  tanw=-|p, 
whence,  equating  the  two  values  of  h  derived  from  the  preceding  expressions, 

li  =  X  tan  A<^  =  S  tan  w, 

h 


or,  as  A<^  is  very  small,  A'A<;f>  =  ^  tan  w,  or,  since  A<f>= -^ 

X  —  o 


hX 

x-s 


=  ^tan 


whence  S  =  h  cot  w  ( ^^^ — ^  ]=h  cot< 


^+x  +  x^  +  - 


whence,  neglecting  the  higher  powers  of  the  fraction  -^r ,  and  putting  for  S  in  the 
expression  its  approximate  value  of  h  cot  w 


S^ll  cot  0) 


(l+^") 


(41) 


90.  As  a  further  example  of  the  use  of  the  equations  derived  in  this  chapter  in 
determining  the  approximate  values  of  the  quantities  concerned,  we  have  the  follow- 
ing: Given  that,  for  the  6"  gun  (m;  =  105,  c  =  0.61),  the  initial  velocity  is  2600  f.  s., 
and  that  the  angle  of  departure  for  a  range  of  5500  yards  is  3°  02'  24";  to  find  the 
approximate  values  of  the  angle  of  fall,  time  of  flight  and  striking  velocity. 

n= ~^  ;   tan  a)  =  tan  d>[2 ;  7=  — -^ —  X  ~ ;  i\= -^ 

gX       '  ^\         nj'  4  7cos<^       '"      3n-l 

y  =  2600    log  3.41497 2  log  6.82994 

2*^  =  6°  04' 48"    -.  ..     sin  9.02491-10 

5^  =  32.2    log  1.50786 colog  8.49214-10 

Z=16500    log  4.21748 colog  5.78252-10 

n  =  1.3474    log  0.12951 


=  0.74215    colog  9.87049-10 


GENERAL  AND  APPROXIMATE  DEDUCTIONS  61 

2-  ^-  =  1.25785   log  0.09963 

n 

3w  =  4.0432 

3n  +  l  =  5.0423    log  0.70263 

3w-l  =  3.0423    ..log  0.48319 colog   9.51681-10 

</>  =  3°  02'  24"    tan  8.72516-10. .     sec  0.00061 

4    log  0.60206 colog  9.39794-10 

Z  =  16500    log  4.31748 

7  =  2600 log  3.41497 colog  6.58503-10.  .     log  3.41497 

3    log  0.30103 


:3°49'19"    tan  8.82479-10 


r  =  8.011    log  0.90368 


r^  =  1709.3    log  3.23281 

(0  =  3°  49'  19". 
r  =  8.011  seconds. 
v^  =  1709.3  f.  s. 

To  find  the  approximate  co-ordinates  of  the  vertex  for  the  trajectory  given  above. 

_yi  +  -dn{n-l)-ly.       _gT^ 
"'  3(n.-l)  '    ^'~~S~ 

n  =  1.3474 
3«  =  4.0422   .  .log  0.60662 
^-1  =  0.3474   ..log  9.54083-10 
3(n-l)  =1.0423    log  0.01795 colog  9.98205-10 

3n(n-l)  =1.4043  ..log  0.14745 

l  +  3n(/i-l)  =2.4043  ..log  0.38099 

Vl  +  3w(n-l)  =1.5506.^  log  0.19050 


VH-3n(n-l) -1  =  0.5506   log  9.74084-10 

X  =  16500    \os  4.21748 


a-o  =  9085    log   3.95832 

T  =  8.011   . .  .log  0.90368 2  log  1.80736 

^  =  32.2    log  1.50786 

8   log  0.90309 colog  9.09691-10 

7/0  =  258.30    log  3.41313 

a;o  =  3028.3  yards. 
2/0  =  358.30  feet. 

For  the  conditions  given  in  the  preceding  problem,  to  find  the  danger  space  for 
a  target  20  feet  high.  There  are  two  formulae  possible,  of  which  the  longer  is  the 
more  exact,  and  is  'the  one  used  in  computing  the  values  of  the  danger  space  for  a 
20-foot  target  given  in  Column  7  of  the  range  tables.  It  should  be  used  whenever 
exactness  is  required.    We  will  compute  by  both  and  compare  the  results. 


;S  =  /icotw         *S'  =  /ieotwfl+ — Y~] 


62  EXTERIOR  BALLISTICS 

n  =  20    log  1.30103 

(0  =  3°  49'  19"   cot  1.17519 


S  =  li  cot  CO  =  299.38    log  2.47623 log  2.47622 log  2.47622 

Z=  16500 log  4.21748 


^^^^=0.0181    log  8.25875-10 

A^^  +1  =  1.0181    log  0.00779 

^  =  304.8   log  2.48402 

By  approximate  formula 99.795  yards. 

By  more  exact  formula 101.600  yards. 

The  variation  in  the  above  more  exact  result  from  the  value  given  in  the  range  table 
is  due  to  the  fact  that  the  value  of  the  angle  of  fall  used  above  is  only  approximate, 
standard  Throughout  this  book,  in  working  sample  problems  showing  the  computation  of 

problem.  ^^^^  ^^^^  ^^^  ^^^  range  tables,  the  work  will  be  done  in  each  case  for  what  will  be 
known  as  the  "  standard  problem  "  of  the  book.  This  will  be  for  a  range  of  10,000 
yards,  for  the  12"  gun  for  which  7  =  2900  f.  s.,  w  =  870  pounds,  and  c  =  0.61.  This  is 
the  gun  for  which  the  range  table  is  Bureau  of  Ordnance  Pamphlet  No.  298;  which 
table  is  given  in  full  in  the  edition  of  the  Range  and  Ballistic  Tables,  printed  for  the 
use  of  midshipmen  in  connection  with  this  text  book.  For  this  gun  and  range,  we 
know,  by  methods  that  will  be  explained  later,  that  the  angle  of  fall,  w,  is  5°  21'  10"; 
therefore,  to  determine  the  danger  space  for  a  target  20  feet  high,  at  the  given  range, 
the  work  for  getting  the  data  in  Column  7  of  the  Range  Table  is  as  follows : 


S  =  h  cot.{l+^-^l^) 


As  we  desire  our  result  in  yards,  however,  we  may  reduce  all  units  of  measurement 
in  the  formula  to  yards,  and  the  expression  then  becomes 


A=6.6667    log  0.82391 log  0.82391 

3 

(0  =  5°  21'  10"    cot  1.02827 cot  1.02827 

R  =  10000    log  4.00000 colog  6.00000  - 10 

-^  cot  w 

o 


R 

h       , 
-—  cot  ( 


=  0.0071    log  7.85218-10 


1+  -^   „       =1.0071    log  0.00307 

6'2o  =  'M.655  yards   log  1.85525 

91.  We  can  now  make  a  comparison  between  the  trajectory  in  vacuum  and  that 
in  air  for  the  same  initial  velocity  and  angle  of  departure. 

Figure  9  represents  on  the  same  scale  the  trajectories  in  air  and  in  vacuum  of  a 
12"  projectile  weighing  870  pounds,  c  =  0.61,  fired  with  an  initial  velocity  of  2900  f.  s., 
at  an  angle  of  departure  of  4°  13.2' ;  the  range  in  vacuum  for  this  angle  of  departure 
being  38315.3  feet  (12771.7  yards),  and  in  air  of  standard  density,  being  30,000 
feet  or  10,000  yards.  In  the  figure  the  ordinates  of  both  curves  are  exaggerated  ten 
times  as  compared  with  the  abscissae,  in  order  that  the  curve  may  be  seen. 


GENERAL  AND  APPROXIMATE  DEDUCTIONS 


63 


If  gravity  did  not  act,  the  projectile  would  move  in  the  tangent  to  the  curve 
0Q^Q2,  and  in  traveling  the  horizontal  distance  x  =  OA,  would  rise  to  the  height 


-4  '  ?3    -'- 

Figure  9. 

Comparison  between  Trajectory  in  Vacuum  and  that  in  Air  for  same  <p  and  same  7.  V. 

AQ.^  =  x  tan  ^.    In  this  case,  assuming  a;  =  20,000  feet  from  the  gun,  we  would  have; 

a;  =  20000    log  4.30103 

<^  =  4°   13.2'    tan  8.86797-10 


gX' 


7/=  1475.7  feet log  3.16900 

The  attraction  of  gravity,  however,  pulls  the  projectile  down  0,P,z=  

while  it  moves  OA  horizontally,  and  so  for  the  ordinate  of  the  trajectory  in  vacuum 
we  have 

AP^  =  y,  =  xtsin  cf>-     J^ 

2  V^  cos^  <^ 

We  have  already  computed  the  value  of  x  tan  ^  as  above  and  found  it  to  be  1475.7 
feet.  Therefore,  computing  the  second  term  of  the  right-hand  member  of  the  above 
equation,  we  have 

g  =  S2:2    log  1.50786 

a;=-. 20000    log  4.30103 Slog  8.60206 

2    log  0.30103 colog  9.69897-10 

«^  =  2900    2  colog  3.07520-10 

<^  =  4°   13.2'    2  sec  0.00236 


^^;^,-^  =   769.93    feet log  2.88645 

a;  tan  <^  =  1475.7     feet 


AP,  =  ,j,  =  ,t.n,-^-^S^,-^  = 


705.8     feet 


64  EXTERIOE  BALLISTICS 

When  the  resistance  of  the  air  also  acts,  retarding  the  motion  of  the  projectile, 
it  takes  longer  for  it  to  move  OA  horizontally,  and  so  gravity  has  longer  to  act,  and 
it  has  fallen  the  further  distance 

and  the  ordinate  of  the  trajectory  in  air  is 

AP'  =  y^  =  x  tan  <^-  ^^-^^  n+ijcx  +  ^Jc'x-) 
■^^  ^      2f-  cos- ^  V     '  3       I  a        / 


or  ?/2  =  a;tan«^-^f 

,  .  ,  72  gij^  2^ 

m  which  n  = .^^-^ 

9^ 


gnx'' 


cos-  ^ 


7  =  2900    , '3  log  6.92480 

2</)  =  8°  26.4' sin  9.16646-10 

^  =  32.2    eolog  8.49214-10 

Z  =  30000    log  4.47712 colog  5.52288-10 


n 


=  1.277 loo-  0.10628 


. 91^ =    log  2.88645 

272cos2«/.  "  . 


_gnx^__  _  983  4   log  2.99273 

27-cos2<^ 

X  tan  </)  =  1475.7 

2/0=   492.3 

In  the  particular  case  represented,  the  projectile  reached  the  ground  after  having 
traveled,  in  air,  the  horizontal  distance  .Y  =  OP"  =  30,000  feet;  but  if  it  had  moved 
in  vacuum  it  would,  at  that  range,  have  been  at  P^,  at  a  height  of  481.2  feet  above  P". 


2 

P2^"  =  a:tan  6—   ^--,/^  ,       (in  vacuum) 
^  27^  cos-  d> 


cos-  <}> 

so  at  a:=  30,000  feet,  we  solve  for  P^P" 

a;  =  30000    log  4.47712 2  log  8.95424 

</)  =  4°  13.2'    tan  8.86797 2  sec  0.00236 

^  =  32.2    log  1.50786 

2    colog  9.69897-10 

7  =  2900    2  colog  3.07520-10 

a;  tan  <^  =  2213.5    log  3.34509 

"-—-  =1732.3   

os^  <^ 

P^P"=   481.2  feet 

Ranare  in  vacuum        ,  •  ^ 

n=     "  ° — -. ; ,  whence  range  m  vacuum  =  wZ 

Itange  m  air 

n  =  1.277    log  0.10628 

Z=30000    log  4.47712 

Range  in  vacuum  =  38315.83    log  4.58340 

The  exaggeration  of  the  ordinates  in  order  to  present  a  serviceable  figure,  as  well 
as  the  arithmetical  results,  shows  how  flat  is  the  trajectory,  according  to  the  most 


GENERAL  AND  APPROXIMATE  DEDUCTIONS 


65 


modern  standards.  It  may  also  be  noted  from  the  figure,  as  we  have  already  pointed 
out,  that  the  vertex  of  the  curve  in  air  is  reached  after  a  shorter  horizontal  travel 
than  is  the  case  for  the  trajectory  in  vacuum. 

* 

EXA:\rPLES. 

1.  From  the  data  given  in  the  first  four  columns  of  the  following  table,  com- 
pute the  value  of  n,  and  thence  the  approximate  values  of  the  angle  of  fall,  time  of 
flight  and  striking  velocity  in  each  case. 


DATA. 

ANSWERS. 

Problem. 

Gun. 
In. 

Initial 

velocity. 

f.  s. 

Angle 
of  departure. 

Range. 
Yds. 

n. 

Angle  of 
fall. 

Time  of 
flight. 

Sees. 

Striking 

velocity. 

f.  S. 

1 

2 

3 

4 

5 

6 
6 
4 
4 
4 
* 

12 
12 
13 

2000 
2900 
2900 
2900 
2900 
2000 
2400 
2250 
2300 

1°  07'  49" 
3     03    51 
1     07    26 
3     03    47 
5     02    58 
1     33    00 
3     33    00 
1     11    00 
1     08    00 

2625 
5394 
2600 
5166 
6599 
1000 
5900 
2000 
2000 

1.308 
1.723 
1.313 
1.798 
2.313 
2.239 
1.249 
1.082 
1.083 

1°  24' 
4     21 

1  24 
4     25 
7     53 

2  24 
4     15 
1     IG 
1     13 

3.34 

8 .  62 
3.32 

8.56 

13.60 

2.90 

8.77 
2 .  83 

2^77 

1984 
1391 
1973 
1320 

977 

700 

1747 

2004 

2046 

6 

7 

8 

9 

*  Small  arm. 


2.  A  6"  gun,  with  2900  f.  s.  initial  velocity,  with  an  angle  of  departure  of 
1°  or  49",  gives  a  range  of  2625  yards,  an  angle  of  fall  of  1°  24'  00"  and  a  time  of 
flight  of  3.34  seconds.  Find  the  coordinates  of  the  vertex  and  the  danger  space  for 
a  target  20  feet  high. 

Answers,     a-^^  1383  yards.     ^0  =  45  feet.     *S'  =  301  yards. 

3.  A  6"  gun  with  2900  f.  s.  initial  velocity,  with  an  angle  of  departure  of 
3°  03'  51",  gives  a  range  of  5394  yards,  a  time,  of  flight  of  8.61  seconds  and  an  angle 
of  fall  of  4°  22'  00".  Find  the  coordinates  of  the  vertex,  also  the  danger  space  for  a 
target  20  feet  high. 

Answers.     a;o  =  2926  yards.    ?/o  =  298  feet.    «S'  =  89  yards. 

4.  The  range  table  of  the  3"  gun  of  2800  f.  s.  initial  velocity  gives  an  angle  of 
departure  of  1°  01'  00"  for  a  range  of  2000  yards,  and  shows  that  the  range  changes 
100  yards  for  each  4'  change  in  the  angle  of  departure.  Find  the  ordinates  of  the 
trajectory  for  that  range  at  points  1700,  1800  and  1900  yards  from  the  gun. 

Answers.     5.93,  4.19,  2.21  yards,  respectively. 

5.  The  angle  of  fall  for  the  2000-yard  trajectory  of  the  3"  gun  is  1°  26'  00". 
Compute  the  danger  space  for  a  target  20  feet  high. 

Answer.     *S'  =  302  yards. 


PART  II. 

PRACTICAL  METHODS. 

IXTEODUCTIOX  TO  PART  II. 

Taking  up  the  more  practical  part  of  the  study  of  exterior  ballistics,  we  here 
find  that  certain  expressions  may  be  derived  that  are  not  equations  to  the  trajectory 
in  air  itself  as  a  whole,  but  which  do  express  with  sufficient  accuracy  for  all  practical 
purposes  certain  relations  that  exist  between  the  values  of  the  several  elements  of  the 
trajectory.  Furthermore,  these  expressions  are  generally  true,  for  all  initial 
velocities,  wherein  they  differ  from  the  equation  to  the  trajectory  in  air  which  we 
have  heretofore  been  using,  which,  as  we  know,  was  only  true  for  initial  velocities 
for  which  Mayevski's  exponent,  a,  was  equal  to  3.  As  was  the  case  in  our  effort  to 
derive  an  equation  to  the  trajectory  in  air,  we  also  find  that,  owing  to  the  differential 
character  of  the  expressions  which  we  are  now  about  to  investigate,  direct  solutions 
are  not  always  possible;  but,  as  was  not  the  case  with  the  equation  to  the  special 
trajectory,  we  will  see  that,  thanks  to  the  ingenuity  of  Major  Siacci,  it  is  possible  to 
obtain  solutions  in  all  cases  by  the  use  of  certain  artificial  mathematical  methods. 
These  methods  are  sufficiently  accurate  for  our  purposes  in  all  cases  and  are  com- 
paratively simple  of  application  after  they  are  understood,  and  are  therefore  entirely 
satisfactory  for  all  practical  purposes. 

We  will  also  see  that,  although  the  mathematical  expressions  involved  contain 
a  number  of  more  or  less  complicated  integral  expressions,  the  working  out  of  which 
for  every  problem  which  we  wish  to  solve  would  be  laborious  in  the  extreme,  this  labor 
has  already  been  performed  for  us  by  different  investigators  of  the  subject,  notably 
Colonel  Ingalls,  and  the  results  tabulated  in  Tables  I  and  II  of  the  Ballistic  Tables; 
and  that,  by  the  use  of  these  tables,  the  labor  of  computation  by  means  of  the  formulae 
in  question  may  be  simplified  to  a  very  great  degree  and  reduced  to  a  minimum. 

The  investigation  of  these  methods  and  the  solution  of  problems  by  them  con- 
stitute Part  II  of  this  book,  and  complete  the  consideration  of  the  trajectory  as  a 
plane  curve. 


CHAPTER  6. 

THE  TIME  AND  SPACE  INTEGRALS;  THE  COMPUTATION  OF  THEIR  VALUES 

FOR  DIFFERENT  VELOCITIES,  AND  THEIR  USE  IN 

APPROXIMATE  SOLUTIONS. 

New  Symbols  Introduced. 

Tu. . .  .Value  of  the  time  integral  in  seconds  for  remaining  velocity  v. 
Ty. . . .  Value  of  the  time  integral  in  seconds  for  initial  velocity  V. 
Sv  ' . .  Value  of  the  space  integral  in  feet  for  remaining  velocity  v. 
Sv  •  •  •  Value  of  the  space  integral  in  feet  for  initial  velocity  V. 

92.  Returning  now  to  Mayevski's  equation  for  the  retardation  due  to  atmos-   Time 

°  -^  ^  integral. 

pheric  resistance,  namely,  -^  =  — W  ^"^  we  see  that  it  may  be  written,  after  sepa- 
ration of  the  variables 

dt=-^-X^  .  (43) 

in  which  A  and  a  are  Mayevski's  constants  and  C  is  the  reduced  ballistic  coefficient 

Suppose  now  a  projectile  to  travel  so  nearly  horizontally  that  its  velocity  is  not 
affected  by  gravitation,  but  is  only  affected  by  air  resistance;  then  the  integration 
of  (42)  between  corresponding  limits,  t.^  and  t^  of  t,  and  v^  and  Vo  of  v,  will  give  the 
elapsed  time  (^2~^i)  corresponding  to  the  loss  of  velocity  (v^  —  Vo).  We  will  assume 
the  velocity  at  the  origin  of  time  {i\)  to  be  3600  f.  s.,  as  that  is  the  upper  limit  of 
initial  velocities  for  which  the  values  of  Mayevski's  constants  have  been  experi- 
mentally determined  (3600  f.  s.  is  also  well  above  all  present-day  service  initial 
velocities,  and  this  point  of  origin  will  therefore  answer  all  practical  purposes  until 
initial  velocities  are  greatly  increased  over  any  present  practice),  and  will  there- 
fore have  ^1  =  0  seconds,  and  Vj^  =  3G00  f.  s.;  and  we  will  designate  by  Tv  the  elapsed 
time  from  the  origin  until  the  velocity  is  reduced  to  v.    Then  we  have 

\''dt=-^\'      ^  (43) 

Jo  ^"^    J  3600      '^ 

by  integrating  which  we  get 

^"^"^  T  ^  a^ (^  ~  (3600)«-ij  (^'^) 

Substituting  in  (44)  the  values  of  A  and  a  given  in  the  first  of  Mayevski's  special 
equations  (14),  we  get 

r.=  c(i?M_4.9.50-2)*  ■  («) 

and  the  value  of  this,  computed  for  any  value  of  v  between  3600  f.  s.  and  2600  f.  s., 
is  the  time  in  seconds  which  it  takes  for  the  air  resistance  to  reduce  the  velocity  of  the 
projectile  whose  ballistic  coefficient  is  C  from  3600  f,  s.  to  v  f.  s.,  when  v  lies  between. 
3600  f.  s.  and  2600  f.  s. 

*  The  number  enclosed  in  brackets  is  the  logarithm  of  the  constant  and  not  the  con- 
stant itself. 


70 


EXTERIOE  BALLISTICS 


Below  2600  f.  s.  the  values  of  A  and  a  change  to  those  given  in  the  second  of 
Mayevski's  special  equations  (14),  and  with  the  new  values  the  limits  of  integration 
change  also,  and  we  have 


dt=- 


whence 


2600 

c 


Tr-T,,,,+    ^     X  a_iLa- 


(2600)"-^ 

which,  after  substituting  the  values  of  A  and  a,  and  that  of  Togoo  as  computed  from 
(45),  reduces  to 

y^^ ^  / [3^^5870]  _ 3_gggQU 

and  the  value  of  this,  computed  for  any  value  of  v  between  2600  f,  s.  and  1800  f.  s.  is 
the  time  in  seconds  it  takes  for  the  air  resistance  to  reduce  the  velocity  of  a  projectile 
whose  ballistic  coefficient  is  C  from  3600  f.  s.  to  v  f,  s.,  when  v  lies  between  2600  f.  s. 
and  1800  f.  s. 

Proceeding  in  a  similar  manner  with  the  other  values  of  A  and  a,  each  integra- 
tion being  performed  between  the  limits  which  correspond  to  the  particular  values 
of  A  and  a  used,  we  obtain  the  following  expressions  for  Tv : 

V  between  3600  f.  s.  and  2600  f.  s. 


T,  =  C  (l^^^^^   -4.9502 


V  between  2600  f.  s.  and  1800  f.  s. 


(46)=-^ 


V  between  1800  f.  s.  and  1370  f.  s. 

T,  =  C  fJAMl^]   -1.8837 
\  V 

V  between  1370  f.  s.  and  1230  f.  s. 

T.  =  C  {^^^  +0.8790) 

V  between  1230  f.  s.  and  970  f.  s. 

r„=e  (^1}^^]  +8.6089) 

V  between  970  f.  s.  and  790  f,  s. 

y.^^^  /J[6J2553]   _i88oi) 

V  between  790  f .  s.  and  0  f .  s, 

r„  =  C  (11:3^]  -18.460) 

Note. — The  above  formulas  give  numerical  values  to  five  places  only.  In  actually 
computing  the  tables,  the  numerical  values  were  carried  out  correctly  to  seven  or  more 
places. 

Space  93.  Multiplying  both  sides  of  equation  (42)  by  v,  we  get  vdt= — j-  X  —^zr, 

in  egra  .    ^^^  putting  ds  for  vdt  in  this,  we  have 

(47) 


ds  = 


A        iJ«-i 


*  The  numbers  enclosed  in  brackets  are  the  logarithms  of  the  constants  and  not  the 
constants  themselves. 


PRACTICAL  METHODS 


71 


and  the  integration  of  this  equation  between  corresponding  limits,  Sj  and  s,  of  s,  and 
i\  and  ^2  of  V,  will  give  the  space  traversed  (s^  —  s■^),  while  the  velocity  is  reduced  by 
the  atmospheric  resistance  from  v^  to  i\,  supposing  again  that  the  path  of  the  pro- 
jectile is  so  nearly  horizontal  that  the  effect  of  gravitation  on  the  velocity  in  the 
trajectory  may  be  neglected.  We  will  assume  that  the  origin  of  space,  as  well  as  the 
origin  of  time,  coincides  with  the  value  of  3600  f.  s.  for  v,  or  8^  =  0  and  rT  =  3600; 
and  we  will  designate  by  Sv  the  space  passed  over  from  the  origin  to  the  point  where 
the  velocity  is  reduced  to  v.    Thus  we  have 

dv 


as= T- 

A 


which  integrates  to 


^      A       a  — 2 


(48) 

r«-2         (3600  )''-•''  ^      ^ 

Substituting  in  (48)  tlie  values  of  A  and  a  given  in  the  first  of  Mayevski's  special 
formulae  (1-i),  we  get 


Sv  =  C {217S0.d -  [2.73774]v°-'^)* 


(49) 


and  the  value  of  this,  computed  for  any  value  of  v  between  3600  f.  s.  and  2600  f.  s., 
is  the  space  in  feet  passed  over  by  a  projectile  whose  ballistic  coefficient  is  C,  while  the 
atmospheric  resistance  reduces  its  velocity  from  3600  f.  s.  to  v  t.  s. 

Proceeding  similarly  for  the  different  values  of  A  and  a  between  the  different 
limits  of  velocity,  exactly  as  was  done  in  the  case  of  the  time  integrals,  we  obtain  the 
following  expressions  for  Si-,: 

V  between  3600  f.  s.  and  2600  f.  s. 

iS\,  =  (7(21780.9- [2.73774]fO--'=) 

V  betwben  2600  f.  s.  and  1800  f.  s. 

*S,  =  C(31227.1- [3.42668]r^-=') 

V  between  1800  f.  s.  and  1370  f.  s. 

,S',  =  C(62875.3-[4.24296]logi;) 

V  between  1370  f.  s.  and  1230  f.  s. 


r  between  1230  f.  s.  and  970  f.  s 


S..=  cfIM12i21 +365.6! 

70  f.  s. 


V  between  970  t.  s.  and  790  t.  s. 

r  between  790  f .  s.  and  0  f.  s. 

,S',  =  C(158436.8-[4.69232]logi;) 

Note. — The  above  formulse  give  numerical  values  to  five  places  only.     In  computing 
the  tables,  the  numerical  values  were  carried  out  correctly  to  seven  or  more  places. 


(50) 


*  The  numbers  enclosed  in  brackets  are  the  logarithms  of  the  constants  and  not  the 
constants  themselves. 


72  EXTEEIOR  BALLISTICS 

94.  Tv  and  Sv  are  known  as  the  "  time  function "  and  "  space  function," 
respectivel}^  and  their  values  have  been  computed  for  values  of  v  from  3600  f.  s.  to 
500  f.  s.,  on  the  supposition  that  C  is  unity,  and  their  values  for  the  different 
velocities  will  be  found  in  Table  I  of  the  Ballistic  Tables  under  the  headings  Tu  and 
Su,  with  the  velocities  in  the  column  headed  «  as  an  argument.  The  reason  for 
representing  the  velocities  in  this  table  by  the  symbol  u  will  be  explained  in  a  subse- 
quent chapter.  To  apply  them  to  any  projectile  for  any  given  atmospheric  conditions 
it  is  only  necessary  to  multiply  the  tabular  values  by  the  particular  value  of  C  for  the 
given  projectile  and  atmospheric  conditions. 
Meaning  of  95.  It  must  be  clearly  borne  in  mind  just  what  the  values  of  T„  and  8u  tabu- 

lated in  these  two  columns  mean ;  that  is,  that  they  are  the  time  elapsed  and  the  space 
covered,  respectively,  while  the  velocity  of  the  projectile  whose  ballistic  coefficient 
is  unity  is  being  reduced  from  3600  f.  s.  to  v  f.  s.  by  the  atmospheric  resistance. 
Therefore,  if  we  wish  to  find  how  long  it  takes  the  atmospheric  resistance  to  reduce 
the  velocity  of  a  projectile  whose  ballistic  coefficient  is  unity  from  f ^  f.  s.  to  v^  f.  s., 
we  would  find  T^^  and  Tv^  from  Table  I,  using  the  values  of  v^  and  v^  as  arguments 
in  the  column  headed  u,  and  then  we  would  have  T  =  Ti,_^  —  Tv^,  and  similarly  for  the 
space  covered  we  would  have  S  =  Sv^  —  Sv^-  If  the  value  of  the  ballistic  coefficient  be 
not  unity,  then  these  two  equations  would  become 

T  =  CiT,-T,J  (51) 

S  =  C{S,-S,J  (52) 

96.  The  above  can  perhaps  be  best  illustrated  by  a  reference  to  Figure  10,  which 
represents  the  complete  trajectory  from  the  point  where  y  =  3600  f.  s.  as  an  origin 
to  its  end,  where  i^^^O.     Now  remembering  the  assumption  that  the  trajectory  is 


7~a7^(y  k5 ; 


Figure  10. 

supposed  to  be  so  flat  that  it  is  practically  a  straight  line,  and  supposing  P^  and  T^ 
are  points  on  it  at  which  the  remaining  velocities  are  those  under  consideration, 
namely,  v^  and  v,,  respectively.  Then  manifestly  from  the  figure,  the  value  of  T  is 
the  difference  between  the  values  of  T^i  ^^d  T^^,  and  the  value  of  8  is  the  difference 
between  the  values  of  8v^  and  8^^,  which  graphic  representation,  with  the  addition 
of  the  ballistic  coefficient,  gives  the  expressions  contained  in  (51)  and  (52). 

Note. — The  curve  shown  in  Figure  10  must  not  be  taken  as  literally  and  mathe- 
matically correct.  It  is  simply  used  to  illustrate  the  point.  While  the  distances  are 
marked  as  both  T  and  S  they  do  not  literally  represent  both  times  and  spaces. 

97.  Suppose  we  wish  to  find  how  long  it  will  take  the  atmospheric  resistance, 
under  standard  conditions,  to  reduce  the  velocity  of  a  6",  105-pound  projectile  from 
its  initial  velocity  of  2600  f.  s.  to  2000  f.  s.,  and  through  what  space  the  projectile 
would  travel  while  its  velocity  is  being  so  reduced,  the  value  of  the  ballistic  coefficient 
being  taken  as  unity.  From  Table  I  we  see  that  the  reduction  from  3600  f.  s.  to 
2600  f.  s.  would  take  0.970  second,  during  which  time  the  projectile  would  travel 
2967.1  feet.    That  is,  ^2000  =  0-970  second  and  /S'2coo  =  2967.1  feet.    Similarly  for  the 


PRACTICAL  METHODS  73 

reduction  from  3600  f.  s.  to  2000  f.  s.  ^,000  =  1-909  seconds  and  *S2ooo  =  5106.1  feet. 

Therefore 

^  =  ^2ooo-r2Goo  =  l-909.-0.970  =  0.939  second,  and 
>S'  =  >S,ooo-^26oo  =  5106.1 -2967.1  =  2139  feet. 

If  the  ballistic  coefficient  were  not  unity,  then  we  W'ould  have  7  =  0.939  X  C  and 
5  =  2139  X  C.  The  variation  from  unity  of  the  ballistic  coefficient  may,  of  course,  be 
caused  by  a  variation  in  any  of  its  factors,  such  as  coefficient  of  form,  density  of 
atmosphere,  etc. 

98.  By  means  of  equations  (51)  and  (52),  if  we  have  given  the  ballistic  coeffi- 
cient, one  of  the  velocities,  and  any  one  of  the  other  quantities,  we  can  find  the  remain- 
ing quantities. 

99.  Suppose  that  a  6"  projectile  weighs  105  pounds;  that  its  coefficient  of  form 
is  0.61 ;  and  that  it  has  an  initial  velocity  of  2562  f .  s. ;  and  that  we  desire  to  know  its 
velocity  after  it  has  traveled  3  seconds,  and  also  how  far  it  will  travel  in  that  time ; 
when  the  barometer  stands  at  30.00"  and  the  thermometer  at  40°  F. 

C-g^;  T=C(T,-T,J,  or  T,  =  ^+T,^;  S  =  C{S,-S,.J 

iv  =  10d    log  2.02119 

8  =  1.056    log  0.02366 colog  9.97634-10 

c  =  0.61    log  9.78533-10.  .colog  0.21467 

fZ-  =  36    W  1.55630 eolos:  8.44370-10 


C=    log  0.65590 colog  9.34410-10 

r  =  3    loff  0.47712 


-^  =  0.66255    log  9.82122-10 

r„  =  1.01840  From  Table  I. 


r,^  =  1.68095  whence. 

^2  =  2122.2  f.s.  From  Table  I. 

Also    <S,,  =  4637.2  From  Table  I. 

5„^  =  3091.6  From  Table  I. 


S,.^-S,=lo-l5.G    log  3.18910 

C=    loff  0.65590 


5^  =  6998.4  feet   log  3.84500 

Therefore  the 

Eemaining  velocity  2122.2  foot-seconds. 

Space  traversed   2332.8  yards. 

100.  Again,  suppose  we  have  a  12"  projectile  weighing  870  pounds,  coefficient 
of  form  0.61,  which  has  a  remaining  velocity  of  2521  f,  s.  after  traveling  4000  yards 
through  air  in  which  the  barometer  stands  at  30.00"  and  the  thermometer  at  68°  F.; 
and  that  we  wish  to  determine  its  initial  velocity  and  time  of  flight  to  that  point. 


74  EXTERIOE  BALLISTICS 

C=  ^  ;  S  =  C{S,-S,:),  whence  *S„  =>S'„,-  f  ;  T  =  C{T,.-T,:) 
bed'  ^ 

w  =  S70    log  3.93952 

S=z0.997    log  9.99870-10.  .colog  0.00130 

c  =  0.61    log  9.78533-10.  .colog  0.21467 

d-  =  U4:    log  2.15836 colog  7.84164-10 

C=    log  0.99713 colog  9.00287-10 

^  =  12000    log  4.07918 

-—  =  1208.0    log  3.08205 

-S',,  =3227.5  From  Table  I. 


5'„,  =2019.5  hence 

t7^  =  2900  f.  s.         From  Table  I. 

Also    T,^  =  1.072  From  Table  I. 

r,,^  =  0.625  From  Table  I. 

2^^,  _  2^^,^-0.447    log  9.65031-10 

C=     log  0.99713 

T  =  4A4:0G    log  0.64744 

Therefore  the  initial  velocity  was  2900  f.  s.,  and  the  elapsed  time  was  4.4406  seconds. 

101.  Again,  suppose  the  projectile  given  in  the  preceding  paragraph  started 
with  an  initial  velocity  of  2900  f.  s.,  and  traveled  for  3  seconds,  under  atmospheric 
conditions  as  given ;  how  far  did  it  go  in  that  time  ? 

C  =  as  before.      T  =  C{  T,,  -  T,, ) .  hence  T,„  =  -^  +  T,^ ;  S  =  C{  S,^_  -  S,^) 

C  =  as  in  preceding  paragraph colog  9.00287  —  10 

T  =  3    log  0.47712 


-^  =  0.302    .■ log  9.47999  - 10 

C  '' 


T,^  =0.625 

From  Table  I. 

r,^  =  0.927 

hence 

v,  =  2635  f.  s. 

From  Table  I. 

Also    ^,,  =  2853.6 

From  Table  I. 

*S,^  =  2019.4 

From  Table  I. 

S^^-S,^=   834.2    log  2.92127 

C=    log  0.99713 

*S'  =  8287    log  3.91840 

Therefore  the  space  traversed  was  2763.3  yards. 

102.  The  foregoing  methods  are  of  course  only  strictly  applicable  to  such  parts 
of  the  trajectory  as  may  without  material  error  be  considered  as  straight  lines,  since 
in  deducing  (51)  and  (52)  we  have  entirely  neglected  the  effect  of  gravitation.  They 
will  give  sufficiently  accurate  results  when  applied  to  any  arc  of  a  trajectory  if  the 
length  of  the  arc  be  not  materially  greater  than  the  length  of  its  chord,  and  if  the 
latter's  inclination  to  the  horizontal  be  not  greater  than  10°  to  15°;  but  they  are 
principally  applied  to  the  entire  trajectories  of  guns  fired  with  angles  of  departure 
not  exceeding  3°  or  4°,  giving  the  striking  velocity  and  time  of  flight  for  ranges  as 
great  as  5000  yards,  in  the  case  of  medium  and  large  guns  of  higb  initial  velocity, 
with  as  much  accuracy  as  is  obtainable  by  any  other  method  of  computation.    It  will 


PEACTICAL  METHODS  75 

be  seen  later  that  these  formulre  may  be  correctly  employed  in  dealing  with  the  pseudo 
velocity,  and  that  the  fornmlaj  then  become  generally  serviceable. 

103.  By  means  of  the  formula^  derived  in  this  chapter  we  may  see  how  to 
determine  by  experimental  firing  the  value  of  the  coefficient  of  form  of  any  given 
projectile.  To  do  this  the  projectile  may  be  fired  through  two  pairs  of  screens  at  a 
known  distance  apart,  and  the  velocity  measured  at  each  pair  of  screens.  Considering 
the  first  pair  of  screens,  which  gave  a  measured  velocity  of  v^,  we  know  that  this 
measured  velocity  is  the  mean  velocity  for  the  distance  between  the  two  screens  of 
the  first  pair,  that  is,  it  is  the  actual  velocity  at  a  point  half  way  between  the  two 
screens  of  the  first  pair;  and  similarly  for  the  second  pair  of  screens.  Therefore  the 
points  of  measurement  giving  the  distance  traversed  while  the  velocity  is  being 
reduced  from  i^^  to  v^  are  the  two  points  half  way  between  the  two  screens  of  each 
pair,  respectively.  It  will  also  be  understood  that  this  distance  between  the  two  pairs 
of  screens  must  be  great  enough  to  furnish  a  material  reduction  in  the  velocity,  but 
not  great  enough  to  violate  the  assumption  on  which  we  have  been  working;  namely, 
that  the  force  of  gravity  does  not  afi^ect  the  flight  of  the  projectile  while  traversing  the 
distance  under  consideration.  Also,  to  avoid  introducing  errors  resulting  from  the 
action  of  gravity,  the  two  pairs  of  screens  should  be  in  the  same  horizontal  plane. 
We  then  have  the  formulae : 

/in 

S  =  CiSv,  —  Sv^),  or,  as  0=.-^—^  >  for  s^^ch  firing 
S  =  ^—.— {S V-,  — S V,) ,  from  which 

from  which  we  can  solve  for  the  value  of  c  from  the  observed  velocities,  by  the  use  of 
Table  I  of  the  Ballistic  Tables. 

104.  As  an  example  of  the  above,  a  Krupp  11.024"  gun  fired  a  projectile  weigh- 
ing 7G0.4  pounds  through  two  pairs  of  screens  328.1  feet  and  6561.7  feet,  respectively, 
from  the  gun,  giving  measured  velocities  of  1694.6  f.  s.  and  1483.3  f.  s.,  respectively, 
at  the  pairs  of  screens.  The  value  of  8  at  the  time  of  firing  being  1.013,  find  the 
value  of  c  for  the  projectile  used.    Using  formula  (53),  we  have 

,«^,.^  =  7389.5  From  Table  I. 

aSi.,  =  6377.3  From  Table  I. 

^,^-.9,,  =1012.2    log  3.00527 

w  =  760.4    log  2.88104 

8  =  1.013    log  0.00561 colog  9.99439-10 

f?  =  11.024    log  1.04234 2  log  2.0SJ68 2  colog  7.91532-10 

/S  =  6233.6    loi?  3.79474 colo<?  6.20526-10 


c=  1.003    log  0.00128 

Actually  this  result  should  have  come  out  c  =  1.00,  as  the  projectile  fired  was  similar 
to  those  actually  used  in  the  experiments  to  determine  the  values  of  A  and  a  for 
standard  projectiles,  but  the  unavoidable  errors  in  the  measurements  of  velocities, 
slight  variations  in  the  projectiles  themselves,  the  efi^ect  of  wind  and  of  the  variable 
density  of  the  air  at  the  difl'erent  points  in  the  path  of  the  projectile,  all  produce 
variations  in  the  experimentally  determined  values  of  c,  so  that  only  the  mean  of  a 
great  number  of  determinations  can  be  safely  considered  as  reliable. 

105.  Again,  for  a  6"  gun,  two  velocities  were  measured,  ^^  =  2550  f.  s.  and 
i;2  =  1667  f.  s.,  at  250  feet  and  5000  yards  from  the  muzzle,  respectively.  If  the 
barometer  stood  at  30.50"  and  the  thermometer  at  0°  F.  at  the  moment  of  firing,  com- 


76  EXTERIOR  BALLISTICS 

pute  the  value  of  the  coefficient  of  form  of  the  projectile  used,  which  weighed  105 
pounds. 

^~  Sd'  ^       S 

5,^  =  6502.1 
*S,.,  =  3131.3 

5,^ -^S.^^  3370.9    log  3.52775 

w  =  105    log  2.02119 

8  =  1.169    log  0.06781 colog  9.93219-10 

d-  =  36    log  1.55630 colog  8.44370-10 

;S'  =  14750    loo-  4.16879 coloff  5.83121-10 


c  =  0.57021    log  9.75604-10 

It  is  to  lie  noted  that  the  distance  between  the  two  points  of  measurement  in  this  case 
is  rather  large  for  our  assumption  in  regard  to  gravity,  and  the  result  is  therefore 
probably  not  very  accurate. 

106.  The  value  of  c  may  also  be  determined  by  a  comparison  of  actual  with 
computed  ranges,  but  the  causes  of  variation  in  the  actual  ranges  are  so  numerous 
that  nothing  less  than  consistent  results  given  by  many  series  of  shots  should  cause 
the  acceptance  as  correct  of  any  value  of  c  thus  determined.  As  a  matter  of  fact  this 
is  the  method  of  determining  the  value  of  c  now  in  use  in  our  navy,  as  explained  in  a 
later  chapter. 

107.  We  have  now  arrived  at  a  point  where  we  may  see  how  to  determine  experi- 
mentally the  initial  or  muzzle  velocity  of  a  projectile,  as  defined  in  paragraph  17  of 
Chapter  1.  To  do  this,  we  actually  measure  its  velocity  at  a  known  distance  from  the 
gun,  and  then,  by  the  formulae  and  methods  given  in  this  chapter,  we  may  compute 
the  velocity  that  complies  with  the  given  definition  of  initial  velocity;  namely,  a 
fictitious  velocity,  not  actually  existent  when  the  projectile  leaves  the  gun,  but  with 
which  the  projectile  would  have  to  be  projected  from  the  muzzle  into  still  air  in  order 
to  describe  its  actual  trajectory. 

Determina-  108.  For  instance,  a  12"  projectile  weighing  850  pounds  was  fired  through  two 

velocity,  screens  100  and  200  feet  from  the  muzzle  of  the  gun,  respectively,  and  the  time  of 
flight  between  the  two  screens  was  0.047  second.  All  conditions  being  standard, 
including  the  coefficient  of  form  of  the  projectile,  compute  the  initial  velocity. 

The  shell  traveled  100  feet  (the  distance  between  the  screens)  in  0.047  second,  there- 
fore the  measured  velocity  was-^r-^  =2127.7  f.  s.,  and  this,  being  the  mean  velocity 

between  the  two  screens,  is  the  actual  velocity  at  a  point  midway  between  them,  that 
is,  at  a  point  150  feet  from  the  muzzle  of  the  gun.  Therefore,  we  have,  yS'  =  150  feet 
and  ^2  =  2127.7  f.  s.,  and  desire  to  compute  t\. 

w  =  850    log  2.92942 

cr-  =  144    log  2.15836 colog  7.84164-10 

0=    log  0.77106 colog  9.22894-10 

>S  =  150    log  2.17609 

■—  =     25.4    log  1.40503 

5...  =4616.5  From  Table  L 


>S'i,^  =  4591.1  whence 


t;i  =  2134.5  f.s.      From  Table  I. 
That  is,  the  initial  velocity  in  this  case  is  2134.5  foot-seconds. 


PEACTICAL  METHODS 


77 


EXAMPLES. 

Note. — The  answers  to  these  problems,  being  obtained  by  the  use  of  formulas  and 
methods  discussed  in  this  chapter,  all  depend  for  their  accuracy  upon  the  correctness  of  the 
assumption  that  in  every  case  the  effect  of  gravity  upon  the  flight  of  the  projectile  is 
negligible  for  the  portion  of  the  trajectory  involved.  They  are,  therefore,  only  accurate 
within  the  limits  imposed  by  this  assumption. 

1.  Given  the  data  in  the  first  five  columns  of  tlie  following  table  and  two 
velocities ;  or  one  velocity  and  either  S  and  T,  compute  the  data  in  the  other  two  of 
the  last  four  columns. 


Projecti 

e. 

Atmos 

phere. 

V,. 

f.s. 

t'o. 

f.'s. 

8. 
Feet. 

T. 

Sees. 

Problem. 

d. 
In. 

to. 
Lbs. 

c. 

Bar. 
In. 

Then 
°F. 

A 

3 
3 
4 
5 
5 
6 
6 
6 
7 
1 

8 
10 
10 
12 
13 
13 
14 
14 

13 

13 

33 

50 

50 

105 

105 

105 

165 

165 

260 

510 

510 

870 

1130 

1130 

1400 

1400 

1.00 
1.00 
0.67 
1.00 
0.61 
0.61 
1.00 
0.61 
1.00 
0.61 
0.61 
1.00 
0.61 
0.61 
1.00 
0.74 
0.70 
0.70 

28.00 
29.00 
30.00 
31.00 
30.00 
29.00 
28.00 
28.25 
29.. 50 
30.75 
31.00 
30.00 
29.00 
28.00 
29.00 
29.53 
30.00 
29.00 

0 
5 
10 
20 
30 
40 
50 
00 
70 
80 
90 
95 
100 
85 
75 
59 
52 
45 

11.50 
2700 
2900 
31.50 
3150 
2600 
2800 
2800 
2700 
2700 
2750 
2700 
2700 
2900 
2000 
2000 
2000 
2600 

935 
2300 
2552 
2003 
2561 
2013 
2247 
2474 
2313 
2133 
2.541 
2114 
2316 
2154 
1833 
1756 
1900 
2343 

3978.0 
1763.3 
3024.2 

6720.9 

5549.4 

9779.2 

5581.0 

5289 . 8 

4471.5 

10799.0 

4562.4 

10942.0 

11920.0 

27530.0 

4746.7 

9042.8 

3947.9 

8767.7 

3.8810 

B 

0.7078 

C 

1.1110 

D 

2.6819 

E 

1 . 9556 

F 

4.2756 

G 

2.22.53 

H 

2.0087 

I 

1.7897 

J 

4.4980 

K 

1.7241 

L 

4.5808 

M 

4.7642 

N 

11.0280 

0 

2.4774 

P 

4.8340 

0 

2.0289 

R 

3.5527 

2.  Given  the  data  contained  in  the  first  seven  columns  of  the  following  table, 
compute  the  value  of  the  coefficient  of  form  of  the  projectile  in  each  case. 


DATA. 

ANSWERS. 

Problem. 

Projectile. 

Value  of 
5. 

Distance  of  pairs  of 
screens  from  gun.* 

Measured  veloci- 
ties at. 

Value  of 
c. 

d. 
In. 

Lbs. 

First 
pair. 
Feet. 

Second 
pair. 
Yds. 

First 
pair. 

f.s. 

Second 
pair, 
f.s. 

A 

3 
3 
4 
5 
5 
6 
6 
6 
7 

13 
13 
33 

50 
50 
105 
105 
105 
165 

1.0.57 
0.989 
l.Ul 
1.062 
0.899 
0.950 
1.107 
1.009 
0.937 

75 
150 
150 
200 
200 
200 
250 
250 
250 

500 

1000 

1500 

1500 

1200 

1200 

i)00 

800 

900 

1100 
2650 
2875 
3130 
31.30 
2550 
2750 
2760 
2650 

1017 
2090 
2390 
2399 
2830 
2348 
2461 
2620 
2448 

1.00570 

B 

1.00120 

C... 

D 

0.672.50 
1.01260 

E 

0.59322 

F 

0.62045 

G 

1.01390 

H 

0.60165 

I 

0.97983 

*  The  distance  given  in  these  columns  is  in  each  case  the  distance  from  the  muzzle 
of  the  gun  to  a  point  midway  between  the  two  screens  composing  the  pair. 


EXTERIOR  BALLISTICS 


3.  Given  the  data  for  actual  firing  contained  in  the  first  six  columns  of  the 
following  table,  compute  the  initial  velocity  of  the  gun  in  each  case. 


DATA. 

ANSWEPvS. 

Projectile 

Di.stance  of  screens 

from  gun. 

Elapsed 

Problem. 

Value  of 
5 

time 
between 

Initial 

velocity. 

d. 
In. 

First 

Second 

screens. 

f.s. 

Lbs. 

c. 

screen. 

screen. 

Sees. 

Feet. 

Feet. 

J 

7 

165 

0.61 

150 

350 

0.925 

0.074 

2716 

K 

8 

260 

0.61 

200 

400 

1.021 

0.074 

2717 

L 

10 

510 

1.00 

250 

450 

0.899 

0.074 

2722 

M 

10 

510 

0.61 

2.50 

450 

1.125 

0.075 

2681 

X 

12 

870 

0.61 

250 

450 

0.937 

0.071 

2828 

0 

13 
13 
14 
14 

1130 
1130 
1400 
1400 

1.00 
0.74 
0.70 
0.70 

250 

250 
250 
250 

550 
550 
600 
600 

1.015 
0.954 
1.115 
0.913 

0.153 
0.149 
0.176 
0.136 

1976 

P 

2024 

0 

2001 

R 

2585 

4.  The  initial  velocity  of  a  3"  13-pound  projectile,  c  =  1.00,  is  2800  f.  s.  Atmos- 
pheric conditions  being  standard,  what  are  the  elapsed  times  until  its  velocity  is 
reduced  to  3600  f.  s.,  2400  f.  s.  and  2200  f.  s.? 

Answers.     0.341,  0.729  and  1.176  seconds. 

5.  The  initial  velocity  of  a  3"  13-pound  projectile,  c  =  1.00,  is  2800  f.  s.  Atmos- 
pheric conditions  being  standard,  what  are  the  spaces  traversed  while  the  velocity 
is  being  reduced  to  2600  f .  s.,  2400  f .  s.  and  2200  f .  s.  ? 

Answers.     922,  1890  and  2917  feet. 

6.  A  12"  projectile  weighing  850  pounds,  c=1.00,  has  an  initial  velocity  of 
2400  f .  s.  What  is  the  remaining  velocity  at  1000,  at  2000  and  at  3000  yards  range ; 
first  in  an  atmosphere  of  standard  density ;  and,  second,  when  the  thermometer  is  at 
82°  F.,  and  the  barometer  at  29.20"? 

Ansivers.     2256,  2118,  1986  f.s. 
2264,  2133,  2008  f.s. 

7.  A  12"  projectile  weighing  850  pounds,  c=1.00,  has  an  initial  velocity  of 
2400  f.  s.  What  is  the  time  of  flight  for  1000,  for  2000  and  for  3000  yards  range; 
when  the  temperature  is  82°  F.,  and  the  barometer  is  at  29.20"? 

Answers.     1.283,  2.647,  4.092  seconds. 

8.  How  long  does  it  take  a  6"  projectile  weighing  100  pounds,  having  an  initial 
velocity  of  2000  f.  s.,  to  travel  1000,  2000  and  3000  yards,  and  what  is  the  remaining 
velocity  at  each  range;  the  temperature  being  44°  F.,  and  the  barometer  at  30.15"? 

Answers.     1.614,  3.492,  5.668  seconds. 
.  1725,    1485,    1282  f.  s. 

9.  A  Krupp  5.91"  gun,  projectile  weighing  112.2  pounds,  having  an  initial 
velocity  of  1667.6  f.  s.,  in  experimental  firing  under  standard  atmospheric  conditions, 
gave  as  a  mean  of  ten  shots,  measured  velocities  of  1656  f.  s.  at  164  feet  from  the  gun, 
and  1358  f.  s.  at  4921  feet  from  the  gun.  Compute  the  velocities  at  those  ranges  and 
from  a  comparison  between  the  observed  and  computed  ranges,  determine  the 
coefficient  of  form  of  the  projectile  used. 

Answers.  Computed  velocities  .  ...  1656.  . .  .  1363  f.  s. 
Observed  velocities  ....  1656 ....  1358  f .  s. 
Value  of  c 1.00 


PRACTICAL  METHODS  79 

10.  A  Knipp  5.91"  gun,  projectile  weighing  112.2  pounds,  having  an  initial 
velocity  of  1763.6  f.  s.,  in  experimental  firing  under  standard  atmospheric  conditions, 
gave  measured  velocities  of  1740.7  f.  s.  at  328  feet  from  the  gun,  and  1369  f.  s.  at 
6562  feet  from  the  gun.  Compute  the  velocities  at  those  ranges,  and  from  a  com- 
parison between  the  observed  and  the  computed  ranges,  determine  the  coefficient  of 
form  of  the  projectile  used. 

Amwers.     Computed  velocities.  .1740 1348  f.  s. 

Observed   velocities..  .1740.7. . .  1369  f.  s. 
Value  of  c 1.00 

11.  A  Krupp  12"  gun,  projectile  weighing  1001  pounds,  having  an  initial 
velocity  of  1721.7  f .  s.,  in  experimental  firing  under  standard  atmospheric  conditions, 
gave  measured  velocities  of  1711  f.  s.  at  328  feet  from  the  gun;  1692  f.  s.  at  984  feet 
from  the  gun,  and  1518  f.  s.  at  6563  feet  from  the  gun.  Compute  the  velocities  at 
those  ranges,  and  from  a  comparison  between  the  observed  and  computed  ranges, 
determine  the  coefficient  of  form  of  the  projectile  used. 

Answers.  Computed  velocities.  .1711. . .  .1690.  . .  .1521  f .  s. 
Observed  velocities..  .1711. ...  1692. ..  .1518  f .  s. 
Value  of  c 1.00 

12.  A  4.72"  gun,  firing  a  projectile  weighing  45  pounds,  c=1.00,  in  experi- 
mental firing,  when  the  thermometer  was  at  65°  F.  and  the  barometer  at  30.43",  gave 
a  measured  velocity  of  2204  f.  s.  at  a  point  175  feet  from  the  muzzle.  What  was  the 
initial  velocity?  Answer.     2228  f.  s. 

13.  A  Krupp  11.024"  gun,  projectile  weighing  760.4  pounds,  in  experimental 
firing,  under  atmospheric  conditi-ons  when  8  =  1.013,  gave  measured  velocities  of 
1746  f.  s.  and  1529  f.  s.  at  points  328  and  6562  feet  from  the  gun,  respectively. 
Com})ute  the  value  of  the  coefficient  of  form  of  the  projectile  used. 

Anstver.     c  =  0.9992. 

14.  A  flat-headed,  6"  projectile,  weighing  70  pounds,  when  fired  through  two 
pairs  of  screens  150  feet  apart,  gave  measured  velocities  at  those  two  points  of 
1881  f.  s.  and  1835  f.  s.  Atmospheric  conditions  being  standard  at  the  time  of  firing, 
compute  the  value  of  the  coefficient  of  form  of  the  projectile. 

Answer.     c=2.46. 

15.  A  Krupp  9.45"  gun,  projectile  weighing  474  pounds,  in  experimental  firing, 
under  atmospheric  conditions  when  8  =  1.06,  gave  measured  velocities  of  1719  f.  s.  and 
1460  f.  s.  at  points  328  feet  and  6562  feet,  respectively,  from  the  gun.  Compute  the 
value  of  the  coefficient  of  form  of  the  projectile  used.  Answer.     c  =  0.9969. 


CHAPTER  7. 


THE  DIFFERENTIAL  EaiTATIONS  GIVING  THE  RELATIONS  BETWEEN  THE 

SEVERAL  ELEMENTS  OF  THE  GENERAL  TRAJECTORY  IN  AIR.    SIACCI'S 

METHOD.     THE  FUNDAMENTAL  BALLISTIC  FORMULA.     THE  COMPU- 

^TATION  OF  THE  DATA  GIVEN  IN  THE  BALLISTIC  TABLES,  AND  THE 

USE  OF  THE  BALLISTIC  TABLES. 

New  Symbols  Introduced. 

u. . .  .  Pseudo  velocity  at  any  point  of  the  trajectory  in  foot-seconds. 

du.  .  .  . Differential  increment  in  u. 

Su-  •  •  •  Value  of  space  function  in  feet  for  pseudo  velocity  u. 

Sy. . . .  Value  of  space  function  in  feet  for  initial  velocity  V. 

Tu.  "  '  Value  of  time  function  in  seconds  for  pseudo  velocity  u. 

Tr.  . . '  Value  of  time  function  in  seconds  for  initial  velocity  V. 

Au.  .  •  .Value  of  altitude  function  for  pseudo  velocity  u. 

Ay.  ■ .  .Value  of  altitude  function  for  initial  velocity  V. 

In-  ■  ■  •  Value  of  inclination  function  for  pseudo  velocity  u. 

Iv.  .  ■  ■  Value  of  inclination  function  for  initial  velocity  V. 
109.  From  the  hypothesis  already  made  in  paragraph  69  that  the  resultant 
atmospheric  resistance  acts  in  the  line  of  the  projectile's  axis,  which  itself  coincides 
with  the  tangent  to  the  trajectory  at  every  point,  it  follows  that  the  trajectory  is  a 
plane  curve.  Por,  if  a  vertical  plane  be  passed  through  the  gun  and  through  any 
point  of  the 'trajectory,  that  plane  will  contain  the  only  two  forces  acting  upon  the 
projectile  while  it  is  at  that  point,  namely,  gravity  and  the  resistance  of  the  air; 
and  so  their  resultant  will  lie  in  that  plane  also,  and  there  will  be  no  force  tending  to 
draw  the  projectile  from  that  plane,  and  so  the  next  consecutive  point  of  the  curve 
must  lie  in  the  same  plane  also ;  and  so  on  to  the  end. 


2t 


Figure  11. 

Derivation  110.  Figure  11  represents  a  portion  of  the  trajectory  with  the  two  forces  acting 

of  the  differ-   ^^^^  ^^^  projcctile,  namely,  its  weight,  lo,  acting  vertically  downward,  and  the 


ential  equa- 
tions. 


resistance  of  the  air,  R=-Xj^v'^,  acting  along  the  tangent  to  the  curve  in  a  direc- 


PEACTICAL  METHODS  81 

f 

tion  opposite  to  that  in  which  the  projectile  is  moving.    Kinetic  equilihrium  results 

from  the  balancina;  of  these  two  forces  bv  the  inertia  forces  — ■  X  -rr  actinsr  in  the 

g        dt 

tangent,  and  —  X  —  acting  in  the  normal,  p  being  the  radius  of  curvature  of  the 

curve  at  the  point  under  consideration.*    Eesolving  forces  along  the  normal  to  the 

traiectorv,  the  inertia  force  —  X  — ,  commonlv  called  the  centrifugal  force,  must 

g       9  ^ 

balance  the  resolved  part  of  w  along  the  same  line,  whence  we  have  —  X  —  =w  cos  6. 

or  — =gcosd  (54) 

P 

In  other  words,  the  acceleration  towards  the  center  of  curvature  is  the  resolved  part 
of  g  in  that  direction.    But  the  radius  of  curvature  p=  — -jr- ,  whence 

v-dO=  —g  COS  6  ds=  —gdx,  or  gdx——v-d9  (55) 

doc 

111.  Dividing  each  side  of  (55)  by  dt  and  putting  Vh  for  -jj,  we  have 

qvh=  —v^  —J- ,  whence   qdt= =  — ;:  , 

^  dt  ^  Vft  i;  cos  ^ 

whence  gdt=  —v  sec  6  dd  (56) 

112.  By  putting  cot  6  dy  for  dx  in  (55)  we  get 

g  cot  6  dy=  —v-d6  or  gdy=  —v-  tan  6  dO  (57) 

113.  By  putting  cos  6  ds  =  dx  in  (55)  we  get 

g  cos  6  ds=  —v'dO  or  gds=  —v-  sec  6  dO  (58) 

114.  Now  resolving  horizontally,  since  the  horizontal  component  of  the  atmos- 
pheric resistance,  —  X  -T7  f "  cos  6,  is  the  only  force  which  acts  to  produce  horizontal 

9         ^ 

acceleration,  -^r--  =  —f^ ,  which  in  this  case  is  negative  acceleration,  we  have 
'  dt~         dt  '  "^ 

dVk  -  d(vcose)  __A^  f, 

dt Jt  C  '  '°'  ^ 

but  from  (56)  we  know  that  gdt=  —v  sec  0  dO,  therefore  the  above  expression  becomes 

gd(v  cos  6)  =~v^<'^^^de  (59) 

115.  Grouping  the  expressions  derived  above  together,  we  have  The  diflfer- 

ential  equa- 

gd(v  cos  6)  =  -^  v^'^^^'dO  (60)  '"°°'- 

gdx=-v~de  (61) 

gdt=-v  sec  Odd  (62) 

gdy=-v- tan  Ode  (63) 

gds=-v- sec  Odd  (64) 

and  these  equations,  (60)  to  (64)  inclusive,  are  the  differential  equations  giving  the 
relations  between  the  several  elements  of  the  trajectory;  and,  could  (60)  be 
integrated,  thus  giving  a  finite  relation  between  v  and  6,  one  of  those  variables  could 

*  See  any  standard  work  on  the  subject  for  the  derivation  of  this  expression. 
6 


EXTERIOR  BALLISTICS 


Pseudo 

velocity. 


be  eliminated  from  equations  (61)  to  (64)  inclusive,  and  then  values  of  x,  y,  t  and  s 
could  all  be  obtained  either  exactly  or  by  quadrature.  (The  "  quadrature  "  method 
need  not  be  taken  up  in  this  elementary  treatise  on  the  subject.) 

116.  Since  we  cannot  integrate  (60),  it  is  necessary  to  resort  to  methods  of 
approximation,  and  a  method  devised  by  Major  F.  Siacci,  of  the  Italian  Army,  has 
been  generally  adopted  by  artillerists  because  of  its  comparative  simplicity  and 
readiness  of  application.  It  is  to  be  noted  that  in  all  the  preceding  chapters  dis- 
cussing the  trajectory  in  air  and  deriving  mathematical  expressions  in  regard  to  it, 
we  have  dealt  with  approximate  methods  only,  and  we  now  see  that  we  are  again,  and 
for  our  final  and  most  approved  methods,  driven  to  fall  back  upon  another  approxi- 
mate system.  However,  this  one,  Siacci's  method,  has  been  found  to  be  accurate 
within  all  necessary  limits  for  ordinary  ballistic  problems,  and  for  our  purposes  we 
may  consider  it  as  exact,  in  contradistinction  to  the  more  approximate  methods  that 
we  have  hitherto  considered.  It  should  not  be  forgotten,  however,  that  the  method  is 
not  literally  exact,  and  that  it  might  be  possible,  under  unusual  and  peculiar  con- 
ditions, that  results  might  vary  appreciably  from  those  actually  existent  in  practice. 
It  is  not  ordinarily  necessary  to  consider  this  point,  but  should  some  unusual  and 
peculiar  problem  present  itself,  it  would  be  necessary  to  consider  whether  the  con- 
ditions were  such  as  to  introduce  material  inaccuracies  into  results  obtained  by  the 
ordinary  methods. 

117.  Taking  rectangular  axes  in  the  plane  of  fire;  origin  at  the  gun;  X,  as 
always,  horizontal,  and  positive  in  the  direction  of  projection;  Y,  as  ahvays,  vertical, 
and  positive  upward;  let  V,  in  Figure  12,  be  the  initial  velocity,  and  <f>  the  angle  of 
departure,  and  let  v  be  the  remaining  velocity  at  any  point,  P,  of  the  trajectory. 
Then,  resolving  v  vertically  and  also  parallel  to  its  original  direction,  and  designating 
by  u  the  component  parallel  to  the  initial  velocity,  we  have 

u  =  v  cosd  sec  (l>  (65) 


JC 


Figure  13. 


118.  The  quantity  u,  which  is  represented  in  Figure  12,  is  known  as  the  "  pseudo 
velocity,"  and  its  use  in  the  solution  of  practical  ballistic  problems  is  due  to  Major 
Siacci,  and  is  the  essence  of  his  method.  It  will  readily  be  seen  that  u=V  at  the 
origin,  where  6  =  cf>,  and  also  at  another  point  in  the  descending  branch  of  the  tra- 
jectory where  6=—4>.    At  the  vertex,  where  6  =  0,  the  pseudo  velocity  differs  most 


PKACTICAL  METHODS  83 

from  the  true  velocity,  its  value  there  being  ii=  v  sec  <f>,  but  if  </>  be  small  the  differ- 
ence is  very  small,  and  for  flat  trajectories  {(f>  not  greater  than  about  5°),  u  may  be 
considered  as  equal  to  v  throughout  the  entire  trajectory  without  material  error. 

119.  Substituting  u  cos  <^  for  v  cos  6  and  ^^   for  v  in  (60),  we  get  siacci's 

cos  u  method. 

gau-    ^    X  ^^g«,,i,^  u       a(f 

whence  ^. -  X  -^  =  -^^  sec^  6  d6  (66) 

A         w<«+i)       cos<"-^^^  ^     ' 

Now  the  value  of  a  definite  integral  remains  unchanged  when  for  any  variable  under 
the  integral  sign  is  substituted  a  constant  which  is  the  mean  value  of  that  variable-^ 

between  the  limits  of  integration.    Thus  for  the  variable  quantity  *  ^  ^,    ,,1  we  may 

substitute  a  constant,  p,  thereby  enabling  {QG)  to  be  integrated  without  introducing 
any  error,  provided  we  assign  to  y8  its  proper  value.  It  may  be  said  here  that,  for  all 
practical  purposes,  in  the  use  of  the  methods  of  exterior  ballistics  in  connection  with 
the  ordinary  problems  of  naval  gunnery,  /8  may  always  be  considered  as  equal  to 
unity  without  exceeding  the  limits  of  accuracy  within  which  we  are  otherwise  able 
to  work.  In  certain  special  classes  of  firing  this  value  of  ^  =  1  cannot  be  used  without 
introducing  material  error,  the  principal  such  classes  being  mortar  and  high-angle 
firing;  but  as  these  are  not  methods  within  the  province  of  ordinary  naval  gunnery, 
the  value  of  ^=1  will  be  adopted  in  the  formulae  throughout  the  rest  of  this  book, 
and  the  factor  ^  will  be  allowed  to  disappear  from  the  formulae,  as,  when  it  is  equal 
to  unity,  it  does  not  affect  the  results  obtained  by  their  use.  (For  a  further  discus- 
sion of  the  value  of  /?,  see  Alger's  Text  Book  on  Exterior  Ballistics,  Edition  of  1906, 
page  58,  et  seq.,  and  other  standard  works  on  the  subject.)     We  know  that 

COS"  <^  =  cos<«--+2'<^  =  cos<»-2)<^  cos-  <^ 
and  by  substituting 

^        C0s'""-'<f>    •  COS'^d)  ,  C0S*<f)  a  2    1 

B=  — ; — ^  m  — ; — ~r  wc  havc  -, — —  =acos^(6 

^      cos"^-^'^         cos<«-^>^  cosc-^'^      ^         ^ 

whence,  by  substitution  in  (6G)  and  expressing  the  direct  integration,  we  get 

NovHf  wecall  — ^-  rT^ny  —^u,  then   (67)  becomes,  after  integration, 

tan</)-tan^=   ^    ^,      (lu-h)   or  tan^  =  tan<i-  ^—^^~    (lu-h)      (68) 
)i  COS"  (f>  2  COS"  (j>  /       \.     / 

120.  Now  let  us  put  ^^l^^f  for  -y  in  (61),  and  we  will  have 

COS  0 

(I dr.  =  —  u-  cos-  ^  sec-  6  (16 
or,  substituting  for  sec-  6  dd  its  value  from  {QQ) 

whence  |%/.r  =  - C £ -1- X  ^^  (69) 

*  The  constant  factor  cos"-2i0  is  taken  out  with  the  variable  factor ^— j-  for  the 

cos"*  —  '  6 

reason  that  their  product,  the  mean  value  of  which  we  call  /3,  differs  very  little  from 
unity. 


84  EXTERIOE  BALLISTICS 

whence  x  =  CiSu-Sr)  (70) 

1     f   du 


in  which  S,,  stands  for  —    . 

A     JM 

121.  Proceeding  in  exactly  the  same  way  with  (62),  we  get 

gdt=  -u  cos  cf>  sec^  e  de= —  X  4-  X  ^ 

['  .,  C     [^  1    ^   du 

or  \   dt= \    -J-  X  -—■ 

Jo  cos<^  iv  A         u'^ 

or  t=:-^{Tu-Ty)  •  (71) 

cos<^ 


in  which  T„  stands  for t-  — -r- 

A  J  w* 


122.  Now,  multiplying   (68)   by  the  differential   of    (70),  and  putting  dij  for 
tan  Q  dx,  we  have 


dxj  —  tan  <^  dx  —  - ^---  (7«  —  ly)  X  ^-  X 


2cos-(^  '  '>  '-  A       m(«-^' 


and  by  putting  — -,- 


f/w 


-^mX  -7^1^  —Au,  this  becomes,  after  integrating. 


C 

2/-a;tan<^=— 5—  [A„-Af-/f('S'«-/Sf)] 

-^  2  cos^  ^  ^  -" 

and,  tinally,  dividing  by  (70),  we  get 

l^=un.t-^-^(i^-iv)  (re) 

X  2cos  <^\Om  — Of  / 

Note  that  in  this  substitution  that  Au  means  the  A  function  of  u,  otherwise  known 
as  the  altitude  function,  this  A  having  of  course  nothing  in  common  with  the  constant 
A  on  the  other  side  of  the  equation. 
The  bauistic  123.  The  formulas  given  iji  equations   (65),   (68),   (70),   (71)   and   (73)   are 

formula.  j,j-^q^^,j^  ^g  ^j^g  ballistic  equations,  and  are  the  ones  on  which  are  based  all  the  principal 
problems  in  exterior  ballistics.  They  are  here  repeated,  grouped,  for  convenience, 
as  follows : 

x=C(Su-Sv)  (73) 

-^  =  tan  <f>-  — ^  fl"^  -ly)  (74) 

tan  ^  =  tan  </>-— ^  (7„-2f)  (75) 

2  cos-  <^ 

t=-^{Tu-Ty)  (76) 

cos  ^ 

v  =  ucos  (l>  secO  (77) 

124.  These  ballistic  formulae  express  the  values  of 

(a)  The  two  coordinates  of  any  point  of  the  trajectory ; 

(b)  The  tangent  of  the  angle  of  inclination  of  the  curve  to  the  horizontal  at  any 
point  of  the  trajectory : 

(c)  The  time  of  flight  to  any  point  of  the  trajectory;  and 

(d)  The  remaining  velocity  at  any  point  in  the  trajectory  directly  as  functions  of 

1.  A  new  variable,  u,  known  as  the  pseudo  velocity,  and  already  defined; 

2.  The  ballistic  coefficient; 

3.  The  angle  of  departure;  and 

4.  The  initial  velocity. 


PRACTICAL  METHODS  85 

The  formula  could  therefore  be  used  in  solving  problems  by  working  out  the 
values  for  each  case  represented  by  the  symbols  S,  A,  I  and  T,  with  their  appropriate 
subscript  letters  in  each  case;  but,  in  order  to  facilitate  the  process,  the  values  of 
these  integrals  corresponding  to  all  necessary  velocities  have  been  worked  out  and 
made  available  in  the  columns  of  Table  I  of  the  Ballistic  Tables.  For  all  velocities 
between  3600  f.  s.  and  500  f.  s.,  using  the  velocities  as  argiiments  in  the  left-hand 
column  of  the  table,  headed  u,  we  may  find  the  corresponding  values  of  the  desired 
integrals,  under  the  appropriate  columns,  headed,  respectively,  Su,  Au,  h  and  Tu. 

1    f"    du    f 

125.  As  shown  bv  (71),  the  value  of  the  definite  integral j-      — —     when   The  time 

''     ^  A.   ]y    Vr      \  function. 

multiplied  liv  —^1  measures  the  time  of  flight  from  a  point  where  the  pseudo 
^  •    cos  </>/ 

velocity  is  V  to  one  where  it  is  u.  We  have  represented  this  definite  integral  by 
Tu  —  Tv,  and  we  require  a  table  from  which  its  value  may  be  taken  for  any  given 
values  of  V  and  u.  But,  as  explained  in  Chapter  6,  the  values  of  Tv{Tu)  in  Table  I 
were  calculated  by  the  use  of  the  formula 

^  _  _1_  f«     _dv_ 

'-^     J  3600      '' 

and  the  integral  given  above  is  the  same  as  this,  except  that  u,  a  velocity,  is  sub- 
stituted for  V,  a  velocity;  so  that  the  results  for  the  same  velocity,  whether  real  or 
pseudo,  would  be  the  same,  and  what  we  formerly  called  Tv  is  the  same  thing  that  we 
now  designate  Tu  —  T^^^q.  Consequently,  we  have  always  that  Tu^  —  Tu=Tv„—Tv^, 
and  the  tabulated  time  functions  may  be  used  indiscriminately  for  either  real  or 
pseudo  velocities,  provided  the  proper  quantity  be  taken  as  an  argument. 

1    f"    du 

126.  In  the  same  way,  in  (70),  the  value  of  the  integral  —  -j-      -j^p^] ,  which   The  space 

we  have  represented  by  Su  —  Sr,  measures  (when  multiplied  by  C)  the  horizontal 
space  traversed  while  the  pseudo  velocity  changes  from  Y  to  u,  and,  as  the  values  of 
Su  in  Table  I,  as  explained  in  Chapter  6,  were  calculated  by  the  use  of  the  formula 

cf  If''        dv 

o,,=  — 


^     .'3600   '^ 

and  the  integral  given  above  is  the  same  as  this,  except  that  u,  a  velocity,  has  been 
substituted  for  v,  a  velocity,  so  that  the  results  for  the  same  velocity,  whether  pseudo 
or  real,  would  be  the  same,  and  what  we  formerly  called  Sv  is  the  same  thing  that  we 
now  designate  Su  —  Ss^qq.  Consequently,  we  now  have  8u^  —  Su^  =  Sv^  —  Sv^,  and  the 
tabulated  values  of  the  space  function,  Su,  may  be  used  indiscriminately  for  either 
real  or  pseudo  velocities,  provided  the  proper  quantity  be  taken  as  an  argument. 

127.  As  shown  by  (68),  the  value  of  the  definite  integral f-  [  when    The  incUna 

A    JyU'-  \  tion  functio 

C       \ 

multiplied  bv    r ^ —     measures  the  change  in  the  tangent  of  the  inclination  of  the 

'■  -^      2  COS^  4>J  or, 

trajectory  from  the  point  where  pseudo  velocity  is  V  to  the  point  where  it  is  u.  We 
have  represented  this  definite  integral  by  lu  —  Iv,  and  we  require  a  table  from  which 
its  value,  for  any  given  values  of  V  and  u  may  be  taken.    To  supply  this,  the  values  of 


'^"-have  been  computed  for  values  of  u  from  3600  f.  s.  to  500  f.  s.,  and 


-19_ 

A.    Joo  u' 

placed  in  Table  I  under  the  heading  7„,  with  u  as  an  argument  in  the  left-hand 
column.  Then,  as  in  the  case  of  the  time  and  space  functions,  we  always  have 
Iu^  —  Iu^  =  Iv^  —  Iv^,  and  these  values  may  be  used  for  either  true  or  pseudo  velocities 
indiscriminately,  provided  the  proper  argument  be  used.    It  would  have  been  equally 


86 


EXTERIOR  BALLISTICS 


well  to  have  tabulated  the  values  of 


__2^r« 


du 


,(a+ 


'—,  making  IggQQ  =  0  instead  of 


■^3600  =  -03138,  as  is  the  case  with  the  integration  performed  as  indicated  for  a  lower 
limit  of  infinity.  This  would  have  made  the  series  of  values  of  the  inclination  func- 
tion, like  those  of  the  space,  time  and  altitude  functions,  begin  at  the  imagined  origin 
where  w=3600  f.  s.,  but  as  we  deal  entirely  with  differences,  the  point  of  origin  is 
immaterial. 

128.  The  equations  from  which  7„  are  computed  are  obtained  by  substituting 
successive  values  of  A  and  a  in 

T  =  _  ll  {  ^^^ 
"  A  Jii<«+i' 

and  integrating,  the  first  integration  being  between  infinity  and  u  (u  from  3600  f.  s. 
to  2600  f.  s.) ;  the  second  between  2600  and  u  {u  from  2600  to  1800)  ;  the  third 
between  1800  and  u  (u  between  1800  and  1370);  and  so  on;  the  results  being  as 
follows : 

u  between  3600  f.  s.  and  2600  f.  s. 

J  _  [4.00897] 
^""  "    u'-'' 

u  between  2600  f.  s.  and  1800  f.  s. 

j^^  [4.48170]  ^Q  QQ,,, 

u  between  1800  f.  s.  and  1370  f.  s. 

h=  [5-38806]  +001776 
ir 

V  between  1370  f.  s.  and  1230  f.  s. 

/«=  [8-35032]  +0  03033 

u  between  1230  f.  s.  and  970  f.  s. 

1^=  [14-30751]  +0.10912 


u  between  970  f.  s.  and  790  f.  s. 


^       [8.55778]  _o.o5028 


(78)^ 


u  between  790  f.  s.  and  0  f.  s. 

/„=  [5-83743]  _o.4i960 

Note. — The  above  formulse  give  numerical  values  correct  to  five  places  only.  In 
computing  the  tables  the  numerical  values  were  carried  out  correctly  to  seven  or  more 
places. 

The  altitude  129.  As  explained  in  paragraph  122,  Au  —  Ay  stands  for  the  value  of  the  definite 


function. 


A 


du 
77^'   we 


1     f"  du 

integral t  \    ^u     (a-i)  >  °^'  substituting  for  /„  its  value  of  — 

A      Jy  W  * 

have 

. 2g_  [«     du 

^'  A^-a  jyu^"""-^^ 

*  The  numbers  enclosed  in  brackets  are  the  logarithms  of  the  constants  and  not  the 
constants  themselves. 


PRACTICAL  METHODS 


87 


and  in  order  that  the  value  of  this  integral  may  be  found  for  any  given  values  of 

2g     f"  (hi 


V  and  u,  the  values  of  — 


have  been  computed  for  values  of  u  from 


3600  f.  s.  to  500  f.  s.,  and  will  be  found  in  Table  I  under  the  heading  Au,  with  values 
of  u  as  arguments  in  the  left-hand  column.  Then,  as  in  the  case  of  the  space,  time 
and  inclination  functions,  we  have  always  Au^—Au^  =  Av^—Av^,  and  the  tabulated 
values  of  the  altitude  function  may  be  used  for  either  real  or  pseudo  velocities  indis- 
criminately, provided  the  proper  argument  be  used. 

130.  The  equations  from  which  the  values  of  Au  are  computed  are  obtained  by 
substituting  the  successive  values  of  A  and  a  in  the  expression 


A  -- J5L 

A-a 


f    du 


and  integrating,  exactly  as  was  done  in  Chapter  6  in  finding  the  values  of  Tv,  except 
that  the  values  of  lu  to  be  substituted  in  the  integral  expression  for  the  value  of 
Au  —  Av  must  include  the  constants  of  integration  whose  values  are  given  in  (78). 
The  results  are : 

u  between  3 GOO  f.  s.  and  2600  f.  s. 

u  between  2600  f.  s.  and  1800  f.  s. 

Au=  LL^-JJll]  _  [1.08179]m''-3- 39.264 
u  between  1800  f.  s.  and  1370  f.  s. 

4„=iMC3^^-[2.49233]logii+ 1052.0 

u  between  1370  f.  s.  and  1230  f.  s. 

^^^[1M6^+  [5.80321],,,, 3^ 


u  between  1230  f.  s.  and  970  f.  s. 
[26.60254] 


[11.75890] 


u  between  970  f.  s.  and  790  f.  s. 

^__^  [^508228]  _  i^Mnil  ^gj^^j 
w*  u 

u  between  790  f.  s.  and  0  f.  s. 

A,=  IM^J^]  +[4.31515]logw-68192.0 

Note. — The  above  formulae  give  numerical  values  correct  to  five  places  only.  In 
computing  the  tables  the  numerical  values  were  carried  out  correctly  to  seven  or  more 
places. 

131.  In  the  preceding  work  in  this  chapter  we  have  followed  the  methods  given 
by  Professor  Alger  in  his  most  excellent  book  on  Exterior  Ballistics,  which  are  the 
generally  accepted  methods,  and  have  derived  certain  ballistic  formulae  as  given  in 


(79) 


*  The  numbers  enclosed  in  brackets  are  the  logarithms  of  the  constants  and  not  the 
constants  themselves. 


Table  I. 


88  EXTEEIOE  BALLISTICS 

equations  (73)  to  (77)  inclusive.  We  have  also  shown  how  the  values  of  the  space 
(Su),  time  (Tu),  inclination  (/«)  and  altitude  (Au)  functions  for  varying  values 
of  the  real  or  pseudo  velocities  have  been  computed  and  made  readily  available  in 
Table  I  of  the  Ballistic  Tables.  Professor  Alger  accepts  the  results  already  obtained 
as  being  sufficient  for  all  practical  purposes,  and  uses  these  equations  in  the  form  in 
which  we  already  have  them  (rearranged  to  suit  each  special  problem)  in  the  solution 
of  ballistic  problems.  Colonel  Ingalls,  however,  proceeded  still  further  with  the 
reduction  of  these  formulae,  and  by  most  noteworthy  mathematical  work  succeeded 
in  getting  resulting  expressions  that  vastly  reduce  the  labor  of  the  computer  below 
that  involved  in  the  use  of  the  formulse  as  they  stand  above.  These  reductions  are 
somewhat  involved,  but,  when  once  carried  through,  so  simplify  the  solutions  of 
problems  and  reduce  the  labor  connected  therewith,  that  Ingalls'  methods  have 
become  generally  accepted  for  work  of  this  nature.  Their  acceptance  and  use  involved 
the  computation  of  another  extensive  table.  Table  II  of  the  Ballistic  Tables,  but  with 
this  table  and  Ingalls'  formulas,  the  work  of  the  computer  is  reduced  to  a  minimum. 
Ingalls'  methods  and  formula  may  be  more  appropriately  considered  in  the  solution 
of  certain  special  problems,  and  the  study  of  them  is  therefore  deferred  to  the  next 
chapter. 
Vse  of  132.  As  an  example  of  the  use  of  Table  I,  let  us  suppose  that  it  is  desired  to  take 

from  it  the  values  of  the  four  functions  corresponding  to  a  velocity,  either  real  or 
pseudo,  of  2727  f.  s. 

For  determining  the  value  of  Su,  we  have:  For  ?i  =  2720,  <S'«  =  2581.1,  and  the 
difference  between  that  value  and  the  value  for  the  next  tabulated  value  of  u,  namely, 
w  =  2740,  is,  by  subtraction  or  as  given  in  the  difference  column  headed  As,  63.4; 
that  is,  a  change  of  20  f .  s.  (at  this  part  of  the  table ;  note  the  reduction  in  the  tabu- 
lated intervals  as  the  velocity  decreases)  in  the  velocity  changes  the  value  of  the 
space  function  63.4.  (Note  that  the  difference  given  in  the  A  columns  is  in  each 
case  that  between  the  value  of  the  function  on  the  same  line  and  the  one  on  the  line 
next  below  it.)     Therefore,  for  ^  =  2727  f.  s.,  we  have 

5„  =  2581.1-^^^^^  =2581.1-22.2  =  2558.9 

or  >g^  =  2517.7+  ^^''t^  ^^  =2517.7  +  41.2  =  2558.9 

and  similarly 

.4„  =  97.94+  ^•^^^^^=97.94  +  1.98  =  99.92 

I„  =  .04791+  -QQQ'^S X 13  ^,04791 +  .00036  =  .04827 

r„  =  .802+  ^^^|^^  =  .802  +  .015  =  .817 

133.  And,  similarly,  suppose  that  we  want  to  find  the  value  of  the  real  or  pseudo 
velocity  (which  one  it  is  depends  upon  the  formula  in  use)  corresponding  to  a  value 
of  /„  =  . 05767. 

The  nearest  tabular  value  to  this  is  /«=. 05775,  corresponding  to  m=2430  f.  s., 
from  which,  by  interpolation, 

w(ort;)=2430+  ^^^=2430  +  2.2  =  2432.2  f.  s. 

134.  If  we  had  desired  to  find  the  value  of  Au  corresponding  to  the  value  of 
7„  =  . 05767  given  above,  we  could  find  u  as  just  described,  and  then  find  the  value  of 
Au  thus 

A„  =  151.46-  '■'''^^■^  =151.46-0.43  =  151.03 


PEACTICAL  METHODS  89 

or  we  could  proceed  without  findino;  the  value  of  u,  thus 

.4„  =  151.46-^^^  X  ^  =  151.46 -0.43  =  1.51.03 

This  latter  method  will  frequently  be  found  necessary  in  the  use  of  Table  IT,  and  it 
should  be  practiced  until  it  can  be  done  quickly  and  accurately. 

135.  As  an  example  of  the  problems  that  may  be  solved  by  the  methods  indicated 
in  this  chapter,  let  us  suppose  we  have  a  3"  gun,  standard  projectile  weighing  13 
pounds,  with  an  initial  velocity  of  2800  f.  s.,  and  that  the  angle  of  departure  for  a 
horizontal  range  of  3000  yards  is  1°  53';  and  that  we  desire  to  determine  the  pseudo 
velocity  and  the  horizontal  distance  traversed  at  the  moment  when  the  projectile  has 
been  in  flight  4  seconds;  atmospheric  conditions  being  standard. 

C=~;    t= -^  (T,-Tv)  ^vhence  Tu=^-^  +Ty;    x  =  OiSu-Sr) 

Cl-  cos  (p  L 

iv  =  rs    log  1.11394 

d-  =  d    log  0.95424 colog  9.04576-10 

C= log  0.15970 colog  9.84030-10 

^  =  4    : log  0.60206 

</>=l°  53' cos  9.99977-10 

^-—^  =  2.768    log  0.44213 

n-^  0.734  From  Table  I. 


T„  =  3.502  whence  w=1411  f.  s.     From  Table  I. 

Su  =  7769.2  From  Table  I. 

»Sy  =  2329.1  From  Table  I. 


6'u-;SV  =  5440.1    log  3.73561 

C= loff  0.15970 


a;=  7858    log  3.89531 

Pseudo  velocity    1411  f.  s. 

Distance   traveled    2619  yards. 

EXAMPLES. 

1.  In  the  4000-yard  trajectory  of  a  3"  gun  (7  =  2800  f.  s.),  <b  =  3°  10', 
a)  =  6°  00',  fo  (velocity  at  vertex)  =1600  f.  s.  and  r„  =  1050  f.  s.  What  are  the 
pseudo  velocities  at  the  three  points  where  the  velocities  are  given  ? 

Answers.     2800,  1602,  1046  f.  s. 

2.  In  the  4000-yard  trajectory  of  a  6"  gun  (7  =  2400  1  s.),  <^  =  2°  52', 
(0  =  4°  09',  t'o  =  1780  f.  s.  and  Va,=  1377  f.  s.  What  are  the  pseudo  velocities  at  the 
point  of  departure,  vertex  and  point  of  fall  ? 

Answers.     2400,  1782,  1375  f.  s. 

3.  In  the  8000-yard  trajectory  of  a  12"  gun  (7  =  2250  f,  s.),  <^  =  6°  23', 
(0  =  8°  58',  fo  =  1690  f.  s.  and  t'a,  =  1347  f.  s.  What  are  the  pseudo  velocities  at  the 
point  of  departure,  vertex  and  point  of  fall? 

Answers.     2250,  1701,  1339  f.  s. 

4.  In  the  10,000-yard  trajectory  of  an  8"  gun  (7  =  2300  f.  s.),  <^  =  11°  00', 
w  =  18°  14'.  ^0  =  1340  f.  s.  and  f a,  =  1036  f.  s.  What  are  the  pseudo  velocities  at  the 
point  of  departure,  vertex  and  point  of  fall  ? 

Answers.     2300,  1365,  1000  f.  s. 


90  EXTERIOE  BALLISTICS 

5.  The  4000-yard  trajectory  of  a  3"  gun  (7  =  2800  f.  s.,  w  =  13  pounds)  has 
(l>  =  3°  10'.  What  is  the  pseudo  velocity  at  a  point  horizontally  distant  from  the  gun 
2000  yards,  and  what  is  the  time  of  flight  to  that  point? 

Anstvers.     1671  f.  s. ;  2.791  seconds. 

6.  The  4000-yard  trajectory  of  a  6"  gun  (7  =  2400  f.  s.,  w  =  100  pounds)  has 
tf>  =  2°  52'.  What  is  the  pseudo  velocity  at  a  point  horizontally  distant  from  the  gun 
2000  yards,  and  what  is  the  time  of  flight  to  that  point? 

Answers.     1829  f.  s.;  2.856  seconds. 

7.  In  the  4000-yard  trajectory  of  a  5"  gun  (7  =  2550  f.  s.,  w  =  50  jwunds), 
^  =  3°  01'.  What  is  the  pseudo  velocity  at  a  point  horizontally  distant  from  the  gun 
2200  yards,  and  what  is  the  inclination  of  the  curve  at  that  point? 

Answers.     1683  f.  s.;  0°  07'  12". 

8.  In  the  6000-yard  trajectory  of  a  6"  gun  (7  =  2300  f.  s.,  w  =  100  pounds), 
^  =  5°  53'.  What  is  the  pseudo  velocity  after  1000  and  after  2000  yards  horizontal 
travel,  and  what  are  the  ordinates  of  the  trajectory  at  those  distances  ? 

Ansivers.     2010  and  1747  f.  s.;  279  and  485  feet. 

9.  In  the  3000-yard  trajectory  of  a  3"  gun  (7  =  2800  f.  s.,  w  =  13  pounds), 
<^  =  1°  53'.  What  are  the  pseudo  velocities  and  what  the  horizontal  distances 
traveled  after  1,  2  and  3  seconds  flight? 

Answers.     2276,  1898  and  1619  f.  s.;  840,  1529  and  2117  yards. 

10.  Determine  the  reduced  ballistic  coefficient  for  a  10",  500-pound  projectile 
of  standard  form,  if  the  temperature  and  barometric  height  at  the  gun  be  84°  F.  and 
29.12",  and  the  time  of  flight  be  16  seconds.  If  the  initial  velocity  be  2000  f .  s.,  given 
the  above  time  of  flight,  find  the  range  and  the  pseudo  velocity  at  the  point  of  fall. 

A7iswers.     C  =  5.432  ;  i2  =  7927  yards;  Wc.=  1151  f.  s. 

11.  Determine  the  reduced  ballistic  coefficient  for  an  8",  250-pound  projectile 
of  standard  form,  the  temperature  being  46°  F.,  and  the  barometer  30.11",  the  time 
of  flight  being  22  seconds.  If  the  initial  velocity  be  2300  f.  s.  and  the  angle  of 
departure  be  11°  00',  find  Uc  from  the  given  value  of  the  time  of  flight,  and  then  find 
the  range.  Ansivers.     C  =  3.85S1;  i/w  =  974.3  f.  s,;  /?  =  9947  yards. 


CHAPTEE  8. 


THE  DERIVATION  AND  USE  OF  SPECIAL  FORMULA  FOR  FINDING  THE 
ANGLE  OF  DEPARTURE,  ANGLE  OF  FALL,  TIME  OF  FLIGHT  AND  STRIK- 
ING VELOCITY  FOR  A  GIVEN  HORIZONTAL  RANGE  AND  INITIAL 
VELOCITY;  THAT  IS,  THE  DATA  CONTAINED  IN  COLUMNS  2,  3,  4  AND  5 
OF  THE  RANGE  TABLES.    INGALLS'  METHODS. 

New  Symbols  Introduced. 

Uu. . . '  Pseudo  velocity  at  the  point  of  fall, 
t'o,. . .  .Eemaining  velocity  at  point  of  fall,  or  striking  velocity. 
Su^ ....  Value  of  the  space  function  for  pseudo  velocity  Uu. 
Tu^ ....  Value  of  the  time  function  for  pseudo  velocity  Uu- 
Aj,^.  . .  .Value  of  the  altitude  function  for  pseudo  velocity  v^a. 
It,^..  . .  .Value  of  the  inclination  function  for  pseudo  velocity  w^. 
Su^. . .  .Value  of  the  space  function  for  pseudo  velocity  u^. 
Tu^. . .  .Value  of  the  time  function  for  pseudo  velocity  u^. 
Au^.  . .  .Value  of  the  altitude  function  for  pseudo  velocity  Uq. 
/„„.  . .  .Value  of  the  inclination  function  for  pseudo  velocity  Uq. 

.  .  Difference  between  two  values  of  the  space  function. 

.  .  Difference  between  two  values  of  the  time  function. 

.  .  Difference  between  two  values  of  the  altitude  function. 

. .  Difference  between  two  values  of  the  inclination  function. 

. .  General  expression  for  value  of  argument  for  Column  1  of  Tabic  TI. 
. .  Special  expression  for  value  of  argument  for  Column  1  of  Table  II. 


z  = 


Z  = 


B'  = 


C, 


C^,  C3, 


AT.. 
AA.. 
A/.. 

X 

x_ 

C  " 

a.  . 

h.. 

a' . . 

t' .  . 

A.. 

B.  . 
A'.. 

rpt 

A  " 
A".. 

etc. 


'General  values  of  Ingalls'  secondary  functions. 


'Special  values  of  Ingalls'  secondary  functions  for  whole  trajectory. 


,  Successive  values  of  C.  The  same  system  of  notation  by  subscripts 
also  applies  for  successive  approximate  values  of  other  quantities 
where  such  use  of  them  is  necessary. 


92  EXTERIOE  BALLISTICS 

136.  As  already  deduced,  the  six  fundamental  ballistic  formulte  are : 

x  =  C{S,-Sr)  (81) 

tan^zz:tan«/>-        ^         {lu-Iv)  (83) 

3  COS"  </> 

t  =  C  seecl>{Tu-Tv)  (84) 

V  =  u  cos  (f)  see  6  (85) 

137.  It  Avill  be  seen  from  the  above  that  special  formulas  may  be  derived  to  fit 
all  particular  cases,  which  special  formulas  will  contain  only  the  quantities  contained 
in  the  above  fundamental  equations ;  that  is,  quantities  that  are  either  known  or  con- 
tained in  the  Ballistic  Tables,  or  the  values  of  which  are  to  be  found. 

Transforma-  138.  Let  US  apply  these  formula  to  the  special  case  under  consideration,  that  is, 

tic  formula'   to  the  derivation  of  special  formulas  for  computing  the  values  shown  in  Columns  2,  3, 

4  and  5  of  the  range  tables.    For  the  complete  horizontal  trajectory  we  may  substitute 

in  the  fundamental  equations  as  given  above,  as  follows:    x=Xj  y  =  ^,  t  =  T,  v  =  V(o 

and  6=  —0).    If  we  make  these  substitutions  we  get: 

From   (80)  C=fi^^  (86) 

From  (81)  S,^  =  Sy+ ^  (87) 


From   (82)  tan  ^=  ,^-^  (4^'^-^--  -^v) 

2cos^cl>\  S,,^-Sy  I 

2sin<^cos-c^^^.^^     ^^Al»,^_^  \  ^gg) 

COS^  \bu^  —  i^V  I 

From   (83)  tan(-a>)  ^tan  </>-       ^        (7^  -7^.) 

2  COS" ^        ^ 

and  substituting  in  this  the  value  of  tan  </>  given  above  we  get 

tan  a>=  ;^-^  (7„^-  i^-~iA  (89) 

2  cos-  ci  V       ^  O,,  ,  — '^F  / 

From   (84)  T  =  C  sec  cl>{T„^-Tv)  (90) 

From   (85)  Via  =  Uu  cos  4>  secot  (91) 

139.  Considering  the  aboye  expressions,  we  may  note  that,  with  the  exception  of 
the  quantities  that  we  desire  to  find,  all  the  quantities  contained  in  them  are  either 
known  or  else  may  be  found  in  the  Ballistic  Tables  (exclusive  of  Table  II).  Pro- 
fessor Alger  uses  them  in  this  form,  in  his  text  book,  for  computing  the  values  of  the 
unknown  quantities  in  those  expressions.  As  an  example  of  his  methods  we  will  noAV 
solve  a  problem  by  the  use  of  the  above  formulae  as  they  stand,  and  without  using 
Taljle  II  of  the  Ballistic  Tables. 

140.  For  the  12"  gun,  7  =  2900  f.  s.,  w  =  870  pounds,  c  =  0.61,  atmosphere  at 
method,    standard  density,  to  compute  the  values  of  the  angle  of  departure,  angle  of  fall,  time 

of  flight,  and  striking  velocity,  for  a  horizontal  range  of  10,000  yards  (without  using 
Table  II)  by  Alger's  method. 


Alger's 


PRACTICAL  METHODS  93 

A„.-A, 


2  cos-  (f>\  "'^        ^u^  —  ^v  I 


PGCw 


We  cannot  get  a  correct  result  without  determining  the  value  of  /;  but  let  us  disre- 
gard that  for  the  moment  nevertheless,  and  proceed  for  the  present  as  though  /=1. 
The  value  of  C  for  standard  conditions  could  be  taken  from  Table  \1,  and  this  will 
usually  be  done  to  save  labor,  but  for  this  first  problem  we  will  compute  it. 

w  =  870    log  2.93953 

c  =  0.61    log  9.78533-10.  .colog  0.31467 

(^2  =  144    log  2.15836 colog  7.84164-10 

C=    log  0.99583 colog  9.00417-10 

2:  =  30000    log  4.47713 

^  =A»S  =3039.0    log  3.48139 

4„^  =  253.05  T„^  =  1.880 

/S„^  =5048.4  j^l-   75,09  Tf  =  0.625  7^  =  0.4388 

Wa,  =  2014.8  _     

Al  =  177.96  Ar  =  1.255 

^A  =  177.96    log  2.25033 

A;S  =  3029    log  3.48129 

M  :^.05875    log  8.76903-10 

A*b 

/r  =  . 04388 


^A 

AS 


-/^  =  .01487    log  8.17231-10 

Ar=  1.255    log  0.09864 

C=    log  0.99583 log  0.99583 

20  =  8°   38'  09" sin  9.16814-10 

(^  =  4°  14'  05" sec  0.00119 

T=  12.464    log  1.09566 

Let  us  now  determine  the  approximate  maximum  ordinate  for  the  above  trajectory. 
To  do  this  we  have 

8 


r=12.464    log  1.09566 2  log  2.19133 

^  =  32.2 log  1.50786 

8   log  0.90309 colog  9.09691-10 

F=  625.3    log  2.79541 

|F  =  416.87  feet,  whence,  from  Table  V,  /=  1.0105. 

We  will  now  repeat  the  preceding  process,  using  the  found  value  of  /  to  correct 
the  original  value  of  C,  in  which  /  was  considered  as  unity,  and  introducing  con- 
secutive subscripts  to  the  several  symbols  to  represent  successive  approximate  found 
values. 


94  EXTERIOR  BALLISTICS 

Ci=    log  0.99583 

/i  =  1.0105    log  0.0045-i 

02=    log  1.00037 colog  8.99963-10 

Z  =  30000 log  4.47712 


A^  =  2997.4    log  3.47675 

<SV  =  2019.4  From  Table  I. 


S^  =5016.8  ^„^  =  250.60  r„^  =  1.864  /„^  =  .07722 

«!=  2023.9  ^y=   ''5.09  7^  =  0.625  /f  =  . 04388 

A.4  =  175.51  Ar=  1.239 

AA  =  175.51    log  2.24430 

A<?  =  2997.4   log  3.47675  7„^  =  .07722 

44  =  .058o5    log  8.76755-10  ^=.05855 

A<S  °  A*S      

,      ^•-  =  -^^  J,  -^=.01867 

^-/^  =  .01467   log  8.16643-10 

/„  -Mzz.01867 log  8.27114-10 

Ar=  1.239   log  0.09307 

C,=    log  1.00037 log  1.00037 log  1.00037 

2 colog  9.69897-10 

«,,  =  2022.9   log  3.30598 

2<^  =  8°26'35"  ..sin  9.16680-10 
(^  =  4°  13'  18"    2  sec  0.00236 sec  0.00118.  .cos  9.99882-10 

a,  =  5°22'00" tan  8.97284-10 sec  0.00191 


r=12.434   log  1.09462  

r„  =  2026.3 log  3.30671 

Results.     c/)  =  4°   13'  IS" 
to  =  5°  22'  00" 
T=  12.434  seconds, 
fw^  2026.3  foot-seconds. 

It  will  be  noted  that  the  two  values  of  T  found  above,  first  by  using  /  as  unity,  and 
second  by  using  the  first  found  value  of  f,  differ  so  very  slightly  that  a  value  of  /  found 
from  the  second  value  of  T  would  not  be  sufficiently  different  from  the  first  to 
materially  affect  the  value  of  the  ballistic  coefficient.  Therefore  we  concluded  that 
the  limit  of  accuracy  of  the  method  had  been  reached,  and  proceeded  to  use  the  values 
already  derived  to  find  the  values  of  the  other  unknown  elements.  For  a  longer 
range  it  would  probably  be  ne('essary  to  repeat  the  work  again,  and  get  a  second,  and 
perhaps  even  a  third  value  of  /. 

141.  The  above  is  the  method  employed  by  Professor  Alger,  and  it  will  be  noted 
that  considerable  mathematical  work  is  required.  Colonel  Ingalls  further  reduced 
the  formula  in  such  a  way  that,  while  the  original  work  of  reduction  of  formulge 
is  much  greater  and  is  somewhat  involved,  nevertheless  the  work  to  be  done  by  the 
computer  in  practical  cases  is  very  much  reduced,  thereby  reducing  the  amount  of 
labor  involved  and  time  expended  in  computing  a  range  table,  as  well  as  reducing  the 
probability  of  error  in  doing  the  work.  He  computed  an  additional  table.  Table  II 
of  the  Ballistic  Tables,  to  assist  in  this.    We  will  now  follow  through  his  method. 


PRACTICAL  METHODS  95 

142.  From  equation   (83)  we  have 

tan  d)  =  tan  ^ -f- —   (lu  —  Iv) 

2  cos-  (^ 

and  if  we  substitute  this  value  in  (83)  we  get 

1-  =tan  6+  ^,-  Ih-  #^1  (02) 

X  2  cos^  <^  \  Su  —  Sv  / 

143.  Let  us   now   introduce   into   the   fundamental   equations    (80)    to    (85)    ingaiis' 

sccondsirv 

inclusive,  and  into  (92)  four  so-called  "  secondary  functions,"  as  follows:  functions 

in  general. 

a=^^i^-Iv  .  (93) 

h  =  Iu-i^^  (94) 

a'  =  a+b  =  Iu  —  Iv  (95) 

t'  =  Tu-Tv  (96) 

The  fundamental  equations  then  become 

From   (SO)  ^'=&  ^^^^ 

From   (81)  x=^C{Su-Sv)  .  (98) 

From   (82)   and   (92) 

JL  =tnn<f>-  -^^  =tan  6+  -^  (99) 

X  2  cos-  (f>  2  COS"  <f} 

From   (83)  tan  e  =  tan  <f>- ^"''^^  ^  (100) 

2  cos-  (f> 

From   (84)  t  =  Cf  sec  cl>  (101) 

From   (85)  t;  =  w  cos  </>  sec  ^  (102) 

144.  Xow  Iniralls'  Ballistic  Tables,  Table  IT,  give  values  for  a,  h,  a'  and  t'  for 

different  values  of  V  and  of  2=  -^ ,  so  that,  in  any  given  problem,  knowing  V  and  x, 

we  can  compute  the  value  of  z  from  the  proper  formula,  and  then  take  the  correspond- 
ing values  of  the  secondary  functions  from  Table  II. 

145.  For  a  complete  horizontal  trajectory,  however,  certain  other  simplifications   secondary 
become  possible,  if  we  have  Table  II  available  for  use.    The  relations  between  <^,  w,   fo^^'ent^re 
T  and  fw,  and  the  other  elements  of  the  trajectory  involve  the  complete  curve  from  the    ^^^^^  °^^' 
gun  to  the  point  of  fall  in  the  same  horizontal  plane  with  the  gun.    Under  these  con- 
ditions we  have  that 

y  =  0;    ^=0;    and^=-a, 
When  this  is  the  case  our  equations  become : 

X  =  CiSu,-Sv)  (104) 

tan  ,^=^-^(4^"— ^-7,) 

2  COS^  0\  bu^^  —  Oy  / 

or,  as  2  tan  ^  cos-  </>  =  2  sin  <f>  cos  <^  =  sin  2<l> 

sin2</>  =  cf'l"^"f^  -ly]  (105) 

V   bu^  —  by  I 

tanco=-tan<^+^^(/„„-M 


96  EXTERIOR  BALLISTICS 

Substituting  in  this  the  value  of  tan  <^  from  the  first  expression  used  in  deducing 
equation  (105)  we  get 

C       fr   *       A,,-Ay 


T  =  C  sec  <j>(T,,^-Ty)  (107) 

t;^  =  ?/„  cos  <^  sec  w  (108) 

146.  Now  let  us  take  another  set  of  secondary  functions,  or  rather  a  set  of 
special  values  of  the  regular  secondary  functions,  as  follows : 

A_^u.-Av_j^  (109) 


Su .  —  5' 


B  =  In^-i'^^-^  (110) 

A'  =  A+B  =  I„^~Iv  (111) 

T'  =  T,^-Tv  (112) 

ingaUs'  147.  ISTow  by  combining  these  special  values  of  the  secondary  functions  given  in 

formula,    equations  (109)  to  (113)  inclusive  with  the  formula?  in  (103)  to  (108)  inclusive, 


we  get 


X  =  C{Su,-Sv)  (114) 

sin2</.  =  AC  (115) 


From  (115)  we  have  that  C=  ^^".^    ,  and  we  also  have  that 

,  BC  ri     2  cos-  (b  tan  w 

tanw=  ^        — or  C  = ^ ■ 

2  cos- ^  B 

,,       J.  sin  2<i      2  cos-  ^  tan  w 

therefore  ■ — ~^  = ^ 

A  B 


whence  we  have  tan  w  =  5' tan  ^  (116) 

B^ 
A 


p 

in  which  B'  =  — t-  ,  and  tabulated  values  of  log  B'  will  be  found  in  Table  II 


T  =  CT' sec  <f>  (117) 

■y^  =  7/^  cos  ^  sec  w  (118) 

148.  It  will  readily  be  seen,  as  already  stated,  that,  as  A,  B,  A'  and  T'  are  only 

special  values  of  a,  h,  a'  and  t',  the  values  of  both  sets  of  secondary  functions,  the 

general  and  the  special,  may  be  taken  from  Table  II,  provided  we  enter  the  table  with 

the  proper  argument  in  each  case,  that  is,  with  the  corresponding  values  of  Y  and  of 

X  X 

z=  ytOt  Z=  -^ ,  z  being  of  course  for  any  point  in  the  trajectory  whose  abscissa  is  x, 

and  Z  being  for  the  entire  horizontal  trajectory,  where  the  abscissa  is  the  range  X. 
And,  as  stated  above,  values  of  log  B  may  also  be  taken  from  Table  II  as  required, 
with  the  same  arguments. 

149.  We  therefore  see  that,  by  certain  somewhat  involved  mathematical  processes. 
Colonel  Ingalls  put  tlie  ballistic  formulae  into  such  form  that  their  use  involves  the 
least  possible  amount  of  logarithmic  work.  By  the  use  of  equations  (113)  to  (118) 
inclusive,  we  can  find  the  value  of  ^,  w,  T  and  i'^,  by  Ingalls'  methods,  provided  we 
have  Table  II  from  which  to  take  the  values  of  the  secondary  functions,  and  provided 


PRACTICAL  METHODS 


97 


we  make  no  correction  for  altitude.  Before  we  proceed  to  computations,  however,  we 
must  find  out  how  to  determine  the  value  of  /,  and,  as  a  preliminary  to  this,  we  must 
consider  the  values  of  the  elements  at  another  important  point  of  the  trajectory,  that 
is,  at  its  highest  point,  summit,  or  vertex.  This  question  is  more  fully  discussed  in  a 
later  chapter,  but  a  certain  preliminary  consideration  of  it  is  necessary  here  in  order 
to  enable  us  to  correct  for  /  in  using  the  formulse  just  derived.  The  coordinates,  etc., 
at  the  vertex  are  denoted  by  the  symbols  already  used,  with  the  subscript  zero. 

150.  At  the  vertex  (Xq,  y^),  we  have  that  6  =  0;  hence  at  this  point  the  funda- 
mental equations  become : 


In^alls' 
formulae 
for  vertex. 


C: 


(119) 


By  putting  ^  =  0  in  the  expression  for  tan  9  we  get 


tan  <^  = 


C 


2  cos-  cj) 


{Iu,-Iv),  or  sin  2cj>  =  C{Iu,-Iv) 


or 


Iu=Iv  + 


sin  2cf> 

C 


cc^  =  C(Su-Sv) 
^=tan<f>--    ^ 


Xq  '      2  COS-  <j>  \  Su^  —  Sy 

t,  =  Cseecf>(Tu,-Tv) 

Vq  =  Ufy    COS    ^ 


-/, 


(120) 

(121) 

(122) 

(123) 
(124) 


151.  Alger  solves  for  the  elements  at  the  vertex  by  the  use  of  the  above  expres- 
sions and  of  Table  I.  Ingalls,  however,  simplifies  the  work  of  the  computer  by  some 
further  preliminary  reductions,  as  he  did  in  the  preceding  case.  The  important  point 
at  issue  in  the  present  problem  is  to  find  a  simple  expression  for  finding  the  value  of 
the  ordinate  of  the  vertex,  in  order  to  find  the  value  of  /.  Ingalls,  by  a  somewhat 
involved  mathematical  process  which  we  will  not  follow  through  in  this  chapter, 
finally  reduced  the  formula  to  give  the  following  expression  for  the  value  of  the 
maximum  ordinate: 

r  =  A"C  tan  (^  (125) 

152.  The  values  of  A"  for  difi'erent  conditions  he  computed  and  tabulated  in  Finding- 

value  of 

Table  II,  with  values  of  Y  and  of  ^^^  ^  as  arguments;  the  latter,  for  reasons  that  '°™ 

will  be  explained  later,  being  taken  as  the  value  of  a\  at  the  vertex.    That  is,  com- 
pute the  value  of  ^^^  ^-  from  the  given  data ;  and  then  use  it,  under  the  proper  page 

in  the  table  for  the  known  value  of  V,  as  an  argument  in  the  A'  column,  to  find  the 
value  of  A"  from  its  own  column.     Then  solve  (125)  to  determine  the  maximum 

ordinate.    A  full  explanation  of  the  reasons  for  using  this  value  of  — j~  as  an  argu- 
ment in  the  A'  column  of  the  table  will  be  given  in  the  next  chapter.* 

*  In  further  explanation  of  this  point  let  us  recall  the  fact  that  we  have  derived  certain 
formulae  relating  to  the  trajectory,  and  that  each  of  these  formulae  has  in  it  one  or  more 
somewhat  involved  integral  expressions.  In  reducing  these  formulae  to  serviceable  shape 
we  replaced  these  integral  expressions  by  certain  symbols,  each  such  symbol  representing 
one  particular  one  of  these  integral  expressions.  Now  by  the  labor  of  previous  investi- 
gators, notably  Ingalls,  the  values  of  each  one  of  these  integral  expressions  has  been 
computed  for  all  possible  useful  conditions  and  the  results  tabulated  in  the  several  columns 
of  the  Ballistic  Tables,  under  the  several  symbols  which  we  used  to  represent  these  integral 
expressions.     (Note  continued  at  foot  of  next  page.) 


98 


EXTERIOR  BALLISTICS 


"Use  of  tables.  153.  Before  proceeding  further,  let  us  now  investigate  the  manner  of  using 

Table  II.  Suppose  we  have  Z  =  2760  and  7  =  1150,  and  we  wish  to  find  the  value  of 
the  secondary  function  A.  Looking  in  the  table  for  y=1150,  and  working  from 
^  =  2700,  we  have,  by  the  ordinary  methods  of  interpolation, 

A -.07643+  -00322x60   _^07643  +  .00193  =  .07836 
Now  sujDpose  we  had  had  Z  =  2700  and  y  =  1175,  we  would  have  had 

^  =  .07643+  ^~'^^t^^^^  ^^^  =:.07643-.00248  =  .07395 
50 

Now  suppose,  to  combine  the  two,  we  had  had  Z  =  2760  and  y=1175,  then  we 
would  have  had 


Interpolation 
formula. 


A       n-r>iQ  ,    .00322x60    ,    ( -.00496)  X 25 
^-.0.643+  —^^ +  ^ ^ 


:  .07643  +  .00193  -  .00248  =  .07588 


Expressing  this  algebraically,  we  would  have  had 

Vt 


A: 


At+^^^ZA+'^^ 


Af.4 


100     ~""  '       \V 
where  At  is  the  next  lower  tabular  value  of  A  below  the  desired  value,  Zt  and  Vt 

Let  us  consider  the  trajectory  OlsiPQ  in  Figure  13.  The  ballistic  formulae  repre- 
sent certain  relations  existing  between  the  elements  of  the  trajectory  at  any  point.  For 
the  point  P,  for  instance,  the  equations  would  contain  certain  integral  expressions  which 
we  have  called  the  "  secondary  functions,"  and  have  represented  by  a,  a',  b,  b'  (of  which 
the  logarithm  is  always  used),  and  t',  and  also  the  pseudo  velocity  u.  Now  for  this  par- 
ticular point,  P,  the  integral  expressions  in  each  of  the  formulse  must  be  integrated  between 
the  proper  limits  for  that  particular  point  and  their  numerical  values  for  this  special  case 
determined;  a  space  integral  being  given  the  limits  x  and  0  for  instance.  Instead  of  hav- 
ing to  actually  perform  this  integration  in  each  case,  however,  Colonel  Ingalls  has  already 

done  the  work  for  us;   and  we  simply  compute  the  value  of  the  argument  z  =  '-^  ,  and 

knowing  V,  we  can  take  from  Table  II  each  of  the  numerical  results  of  integration  between 
the  proper  limits  that  we  desire.  So  for  each  point  of  the  curve  there  is  a  special  numerical 
value  of  each  of  the  integrals  represented  by  the  symbols  a,  a',  etc. 


Figure  13. 


In  a  subsequent  chapter  we  will  use  values  so  found  for  different  points  of  the  curve, 
but  for  the  present  we  are  dealing  with  the  entire  trajectory,  and  want  to  find  the  values 
of  the  functions  for  the  point  of  fall,  Q.  Therefore,  x  becomes  X,  the  range,  and  our 
integral  expressions  are  represented  by  A,  A',  B,  B'  and  T',  and  u  becomes  «jj.    Knowing 

V,  we  therefore  compute  the  value  of  Z  = -^  ,  and  thence  from  Table  II  we  may  find  the 

correct  values  of  each  of  the  integral  expressions  represented  by  the  symbols  A,  A',  etc., 
and  also  of  the  pseudo  velocity  at  the  point  of  fall,  xi^.  Keep  it  firmly  in  mind  that  these 
particular  symbols  and  values.  A,  A',  etc.,  are  only  special  symbols  and  values  of  the  gen- 
eral symbols  a,  a',  etc.,  at  one  particular  point  of  the  trajectory,  that  is,  at  the  point  of  fall, 
or,  expressed  in  another  way,  that  they  pertain  to  the  entire  trajectory  and  not  to  any  part 
thereof  or  to  any  other  point  of  the  curve  than  the  point  of  fall.  (Note  continued  at  foot 
of  next  page.) 


PEACTICAL  METHODS  99 

are  the  next  lower  tabular  values  of  Z  and  Y  below  the  given  values ;  AT  is  the  differ- 
ence between  successive  tabular  values  of  F  (that  is,  50  f.  s.  for  the  table  7=1150  to 
1200  f,  s.,  and  100  f.  s.  for  all  other  tables  given  in  the  tables  to  be  used  with  this 
book).  \Z  is  always  100  for  all  tables,  and  it  is  therefore  allowed  to  remain  in 
numerical  form  in  the  above  algebraic  expression,  ^.za  and  Atm  are  the  differences 
given  in  the  proper  line  of  the  A  columns  pertaining  to  A.  Care  must  be  taken  to  use 
the  proper  signs  for  all  the  quantities  given  in  the  above  expression.  It  will  also  be 
seen  at  once  that  any  other  one  of  the  secondary  functions  may  be  substituted  for  A 
in  the  above  expression  provided  we  exercise  due  care  in  regard  to  the  signs.  It  must 
also  be  noted  that  the  formula  applies  for  the  next  lower  tabular  values  only.  If  we 
work  from  the  nearest  tabular  values,  there  would  be  a  change  of  signs  in  the  expres- 
sion if  the  nearest  tabular  value  happens  to  be  the  next  higher  tabular  value  in  any 
case. 


Similarly,  for  the  vertex,  where  x  =  Xo,  we  have  another  set  of  special  values,  but  here 
another  symbol  (which  also  represents  an  integral  expression)  A"  has  been  introduced. 
So  we  would  have  as  our  special  symbols  for  the  vertex  Co,  a./,  A",  6o,  W  and  U,  and  the 
pseudo  velocity  i<o.  Remember  that  A"  pertains  exclusively  to  the  vertex,  and  to  no  other 
point  of  the  curve.  If  we  only  knew  the  value  of  Xr,,  we  could  find  the  numerical  values  of 
each  of  the  symbols  for  the  vertex  (each  an  integral  expression  integrated  for  the  proper 

limits  for  the  vertex)  by  computing  the  value  of  Zo=  -^,  and  thence  v/ith  that  value  and  the 

value  of  y  as  arguments  taking  them  from  Table  II  just  as  for  any  other  case.  But  ice  do 
not  ktiow  the  value  of  Xo.  Fortunately,  however,  it  happens  to  be  a  fact  (susceptible  of 
mathematical  proof,  as  will  be  shown  in  a  later  chapter)  that  the  value  of  A  for  the  point 
of  fall  is  always  numerically  equal  to  the  value  of  c^,'  for  the  vertex.  Now  we  have  already 
a  method  of  finding  the  value  of  A  for  the  point  of  fall  when  we  know  the  range  and  the 
initial  velocity,  and  having  found  that  value  of  A,  we  know  that,  because  of  the  coinci- 
dence stated  above,  we  have  also  the  value  of  Co'  for  the  vertex,  for  the  values  of  these  two 
functions  are  numerically  the  same. 

Now  we  are  trying  to  find  the  value  of  A",  an  integral  expression  relating  solely  to  the 
vertex,  and  from  what  has  been  said  above  we  know  that  we  have  found  the  value  of 
another  function  of  the  vertex;  namely,  a.,'.  Therefore,  with  our  found  value  of  «„'  in  the 
A'  column  of  the  table,  we  interpolate  across  to  the  A"  column  to  find  the  proper  value  of 
A"  for  substitution  in  the  formula  Y  =  j/,,  =  A"C  tan  0. 

X 

The  question  has  been  frequently  asked  why,  given  Z  =  -^^  ,  we  cannot  take  out  the 

corresponding  value  of  A"  direct.  The  answer  to  this  question  is  that  Z  is  for  the  'point 
of  fall  for  the  entire  trajectory,  only ;  therefore,  if  we  do  this  we  will  get  the  value  of  A" 
corresponding,  not  to  our  vertex  N,  but  to  the  vertex  of  another  trajectory  entirely,  that  is, 
of  the  trajectory  ON'Q',  which  vertex  lies  vertically  over  the  point  Q,  the  point  of  fall  of 
the  trajectory  with  which  we  are  working. 

The  value  of  z^  for  our  correct  trajectory  may  be  found  from  our  found  value  of  Qq' 
by  interpolating  back  from  it  in  the  A'  column  to  the  Z  column  by  the  use  of  equation 
number  (128) ;  and  having  found  this  we  may  then  find  the  value  of  x,,  from  the  expression 

Xo  =  ZaC  (  as  Zo  =  -/^"  )  >  which  is  a  process  employed  in  a  later  chapter.     From  this  found 

value  of  Zo  we  may  also  take  from  the  tables  the  values  of  the  other  secondary  functions 
for  the  vertex,  a^,  &o.  log  be,'  and  t,/,  and  of  the  pseudo  velocity  at  the  vertex,  u,,.  But  to  get 
the  correct  value  of  A"  it  must  be  taken  out  by  cross  interpolation  as  already  described, 
and  not  by  finding  first  0,,  and  then  working  with  a  found  value  of  Zn  as  an  argument. 

There  are  points  in  the  explanation  contained  in  this  foot-note  which  are  perhaps 
not  mathematically  perfect,  but  it  is  hoped  that  these  explanations  will  nevertheless 
lead  the  student  to  a  better  practical  understanding  of  the  reason  why,  when  we  have 
found  a  numerical  value  for  A  for  the  whole  trajectory,  we  go  over  with  it  to  the  A'  column 
when  we  wish  to  interpolate  to  get  the  value  of  the  integral  expression  A"  and  that  of 

So  =  Y^  ,  both  of  which  values  pertain  solely  to  the  vertex. 


100  EXTERIOE  BALLISTICS 

Interpolation  Eepeating  the  expression  just  derived,  and  also  solving  it  for  Z  and  Y ,  we  get 

formulae. 


AF 

100 


Z-Zt  ,     1 


U--4.)-^^*Ar.4]  (128)* 


Az 

From  the  first  of  these  we  find  the  value  of  A,  given  Z  and  Y  (or  of  any  other 
of  the  secondary  functions  in  place  of  A).  From  (127)  we  can  find  Y ,  and  from 
(128)  we  can  find  Z,  given  Z  or  Y  respectively,  and  A.  or  any  other  secondary  func- 
tion in  its  place.  The  expressions  of  course  simplify  greatly  when  working  with  a 
tabular  value  of  either  Z  or  Y ,  in  which  case  Z  —  Zt  or  Y  —Yt  becomes  zero. 
Cross  inter-  154.  In  practice  it  is  often  necessary  to  find  the  value  of  A"  corresponding  to  a 

formula  found  value  of  A',  knowing  Y ,  in  finding  the  value  of  the  maximum  ordinate,  as  will 
^^A"  be  shown  later,  under  circumstances  when  we  do  not  care  to  know  the  value  of  Z  or 
of  any  other  of  the  secondary  functions.  To  do  this,  that  is,  to  find  the  double  inter- 
polation formula  for  crossing  direct  from  A'  to  A"  without  finding  Z,  knowing  Y , 
substitute  in  (126)  expressed  for  A",  the  value  of  Z  found  from  (128)  expressed  for 
A' .    The  resultant  expression  is 

A"  =  A/'-i^X^^^^-^-t-^^f^Ar^<,+  (l'-A/)^'      (129)* 
AK  Az^'  Ak  i^zA' 

If  we  are  working  with  a  tabular  value  of  Y ,  which  is  fortunately  generally  the 
case,  then  Y  —Yt  —  ^,  and  the  above  formula  simplifies  greatly,  becoming 

A"  =  A/'+(A'-A/)4^^  (130)* 

As  examples,  suppose  we  have  7  =  1175  and  Z  =  2760,  and  desire  to  find  the 
corresponding  values  of  the  secondary  functions.  Then  7— 7^  =  25  and  Z  — Zi  =  60, 
and  we  have 

A=  07G43-h  •QQ^^^X'^Q  +  (-•0Q^Q'3)x25  _,07643-f-. 00193 -.00248  =  .07588 

.0072X60         ( -.0096)  X 25  ^.i63l  +  .00432-.0048  =  .16262 

A"  =  1435-F^^^  4- ^^  =  1435  +  33  +  4  =  1472 
100  oO 

B-  0866+  -Q^"^^^^  +  (--00^6)  X 25  ^,0866  +  . 0024 -.0023  =  .0867 

locr5'=  05465+  •^^^^^^'^^  +  -00522x25  ^,05465  +  .00089  +  .00261  =  .05815 
»         ■  100  oO 

^  =  948.5+ i^;?)-^^^  _^  22J3X25  ^948.5-3.18  +  11.3  =  956.62 
100  0^ 

y,_    ^3      .106x60        (-.08)x25  ^2.613  +  .0636-.04  =  2.6366 

7)'-117+  122160        1^6)  X 25  ^117  +  6-3  =  120 
^        ^      100  50 


*  It  must  be  noted  that  the  interpolation  formulae  here  derived  for  use  with  Table  II 
of  the  Ballistic  Tables,  neglect  second  and  higher  differences.  They  therefore  give  results 
that  are  accurate  only  within  certain  limits,  which  limits  are  sufficiently  narrow  to 
permit  the  formulae  to  be  used  for  the  purposes  for  which  employed  in  this  text  book. 
A  caution  must  be  given,  however,  against  using  them  for  other  purposes  without  ascertain- 
ing whether  or  not  they  will  give  sufficiently  accurate  results  for  the  purpose  in  view. 
A  case  where  they  cannot  be  successfully  used  is  given  in  Chapter  14.  (See  foot-note  to 
paragraph  239,  Chapter  14.) 


PRACTICAL  :\[ETHODS  101 

155.  Suppose  we  have  .1"  =  14T2,  as  given  above,  and  know  T  =  11T5,  and  wish 
to  find  the  value  of  Z.    From  (1'38) 

Z  =  2:00+  ^  f37-  ^-15)  =2:00+  34^  =2760 
55    \  oO     /  00 

Suppose  we  had  been  given  w  =  956.G2  and  Z  =  2760,  and  wanted  to  find  the  value 
of  Y ,  knowing  it  to  be  between  1150  and  1200  f.  s.    From  (127) 

And  we  could  proceed  similarly  with  the  other  secondary  functions. 

N"ow  suppose  that  we  know  that  7  =  2775  and  that  A' =  .20235,  and  desire  to 
find  the  corresponding  value  of  A" .    Then  from  (129)  we  have 

,„_,oo-       75^  ^  (-.0143)  X  74   ,    75x11    ,    .00225x74 
A   -4Joo-  ^^^  x  -p^„^  +  ^^^  +  ^.0^56~ 

^"  =  4935  + 141.7  + 8.3 +  29.7  =  5114.7 

Suppose  we  find  the  value  of  A"  from  the  above  data  by  first  finding  the  value 
of  Z  by  (128)  and  then  of  A"  from  that  by  (126)  expressed  for  A" . 

Z  =  7G00+-^?;;  f. 00225-  ^~ '^^tfl  ^'^'^l  =7600  +  231.7  =  7831.7 
.OO06  \  100  / 

,„     .^Q.,  ,   31.7x73    ,    13x75       ,.^.  ^ 

-^  =^^^^''^+-^00- +  -100-=^^^^-^ 

which  shows  that  the  results  obtained  by  the  two  methods  difl^er  slightly.     To  get 
results  to  coincide  with  those  obtained  in  this  text  book  (129)  must  be  used.* 

156.  Eeturning  now  to  our  formulfB,  and  having  found  an  expression  for  the 
value  of  the  maximum  ordinate,  on  the  basis  that  the  mean  density  of  the  air  through 
which  the  projectile  travels  is  the  same  as  that  at  a  point  two-thirds  the  maximum 
ordinate  above  the  gun  (which  is  absolutely  true  for  a  uniformly  varying  atmosphere 
and  for  a  trajectory  that  is  a  true  parabola)  ;  and  starting  Avith  Chauvenet's  dis- 
cussion of  atmospheric  refraction,  Colonel  Ingalls,  by  another  rather  involved  mathe- 

*  It  must  be  noticed  here,  as  well  as  elsewhere  throughout  this  text  book,  that  interpo- 
lations are  carried  out  to  more  decimal  places  than  is  strictly  justifiable  for  the  limits  of 
accuracy  obtainable  with  the  ballistic  and  logarithmic  tables  used.  The  rule  adopted 
for  instruction  of. midshipmen  has  been  to  have  them  carry  out  all  interpolations  to  five 
working  decimal  places,  as  five  place  logarithmic  tables  are  standard  at  the  Naval 
Academy;  thus  2534.7,  2.5347,  .0025347,  etc.  This  has  been  found  advisable,  not  because 
it  is  expected  to  attain  more  accurate  results  thereby,  but  in  order  that  a  comparison  of 
the  relative  accuracy  of  the  tabular  and  logarithmic  work  of  all  midshipmen  may  be 
secured.  Due  consideration  was  given  to  all  possible  methods  of  attaining  this  object, 
and  the  only  available  rule  that  could  be  given  them  that  would  not  depend  upon  individual 
judgment  in  any  case  and  yet  that  would  fit  all  cases  was  the  one  that  is  set  forth  above. 
Under  it,  with  a  given  set  of  data,  all  midshipmen  should  secure  the  same  results  (within 
decimal  differences  in  the  last  place),  and  therefore  their  results  can  be  compared  and 
relative  marks  fairly  awarded.  In  service,  however,  this  practice  would  simply  be  an 
apparent  effort  to  attain  an  impossible  degree  of  accuracy,  which  might  in  some  cases, 
indeed,  introduce  small  errors  into  the  work  that  would  otherwise  be  avoided  by  a  less 
rigorous  rule  for  interpolations.  Only  an  experienced  mathematical  judgment  can  tell  to 
what  degree  each  process  of  interpolation  should  be  carried  in  each  separate  case,  and 
this  is  not  possessed  by  midshipmen.  Some  simple,  absolute  rule  was  therefore  necessary 
for  them,  and  this  being  a  simple,  invariable  rule,  it  is  believed  that  the  end  justifies  the 
means  in  this  particular.  This  note  should  be  a  caution  against  attempts  in  service  to 
attain  impossible  accuracy  of  results  by  excessive  carrying  out  of  interpolations. 


102  EXTERIOR  BALLISTICS 

matical  process  which  it  is  unnecessary  to  follow  through,  derives  an  equation  for 
determining  the  value  of  /  as  follows : 

loglog/  =  logF  + 5.01765 -10  (131) 

ingaUs'  soiu-  157.  With  the  aid  of  the  formulae  of  Colonel  Ingalls  already  given,  we  may  now 

cessive^ap'-    proceed  to  the  solution  of  the  original  problem  already  solved  by  Alger's  method, 

proxima  ions.    ^^  g^yg^  jj^  paragraph  140,  by  the  methods  used  in  preparing  the  range  tables  by  the 


expert  computers  of  the  Bureau  of  Ordnance.    The  data  is  the  same  as  before. 

— y-  ;    argument  for  Table  II,  Z  —  -^ 
cd-  0 


C^=  -I'L  .    argument  for  Table  U,  Z  =  ^;  sin  2ct>  =  A0 


wz=870    log  2.93952 

f  =  0.61    log  9.78533-10.  .colog  0.21467 

d-  =  lU    log  2.15836 ..colog  7.84164-10 


C,=    log  0.99583 colog  9.00417-10 

Z  =  30000   loo-  4.47712 


^,  =  3029    log  3.48129 

For  7  =  2900  (table  for  7  =  2900  to  3000),  for  Z  =  3029,  we  have  Ai  =  .014882 

/I  =  .014882    log  8.17266-10 

Ci=    log  0.99583 

2<^i  =  8°  28'  34"    sin  9.16849-10 

^1  =  4°   14'  17".  ..  .First  approximation,  disregarding /. 

Having  obtained  the  preceding  approximation  to  the  value  of  6,  we  may  now  proceed 
to  determine  a  second  ap]oroximation  to  the  value  of  the  same  quantity,  and  this  time 
we  can  correct  for  a  value  of  /,  using  equations  (125)  and  (131). 

Y  =  A"C  tan  4>         loglog/  =  log  F+ 5.01765 -10 

For  finding  A"  from  Table  II  we  use  a^'^    sm^</)  __()][4882^  as  already  deter- 
mined, as  an  argument,  in  its  proper  column.    For  this  the  table  gives  for 
A' =  .0148     AzA'  =  .0012     A"  =  849     A;j.i"  =  57 

,.       n                                   ^"     Qioi    .000082x57      Q..-,  Q 
therefore  A^  =849H ^-— ^ =8o2.9 

A/'  =  852.9    log  2.93090 

C^=    log  0.99583 

<^i  =  4°   14'  17"   tan  8.86983-10 

J\=    log  2.79656 

Constant    log  5.01765-10 

/j=    log  0.00652 log-log  7.81465-10 

C,=    log  0.99583 

C,=    log  1.00235 colog  8.99765-10 

Z  =  30000    log  4.47712 

Z,  =  2983.8   log  3.47477 


PRACTICAL  METHODS  103 

From  Table  II,  4,  =  .014599 

4.  =  .014599    log  8.16432  - 10 

C\=    log  1.00235 

2<^,  =  8°  26'  26"    sin  9.16667-10 

</>o  =  4°   13'  13". ..  .Second  approximation. 

Having  obtained  the  above  second  approximation  to  the  value  of  ^,  we  see  that  it 
differs  from  the  first  approximation  by  over  one  minute  of  arc,  and  we  therefore 
cannot  assume  that  the  second  value  is  sufficiently  accurate.  We  therefore  repeat  the 
above  process  to  get  a  third  approximation. 

From  above  work  ^2  =  2983. 8  and  Ao  =  ao/  =  . 014559,  which  gives,  from  Table  II, 

A;'  =  793+  :000899x56  ^g3g_^ 

^/'  =  838.7   log  2.92360 

C„=    log  1.00235 

0^  =  4°   13'  13"    tan  8.86800-10 

J\_=    log  2.79395 

Constant    log  5.01765-10 

/.,=    log  0.00648 loglog  7.81160-10 

C\=    log  0.99583 

C,=    log  1.00231 colog  8.99769-10 

X=z 30000    log  4.47712 

^3  =  2984.1   log  3.47481 

Hence  from  Table  II,  as  before, 

A3  =  .014601 

A3  =  .014601    log  8.16438-10 

C,=    log  1.00231 

2</)3  =  8°  26'  28"   sin  9.16669-10 

<^3  =  4°   13'  14".  ..  .Third  approximation. 

This  value  is  only  one  second  in  value  different  from  the  second  approximation,  and 
is  therefore  practically  correct;  but  we  see  that  there  is  still  a  small  difference  between 
the  last  two  successive  values  of  C,  and  as  we  desire  to  determine  accurately  and 
definitely  a  correct  value  of  C  for  use  in  further  work,  we  will  proceed  with  another 
approximation. 

A3  =  .014601 

.„     ^^o  ,    .000901x56  _QQQo 

^^  ='^^+      .0011      -^^^'^ 

A/'  =  S3S.9   log  2.92371 

C,=    log  1.00231 

</.3  =  4°   13'  14"    tan  8.86803-10 

1-3=    log  2.79405 

Constant    log  5.01765-10 

f^=    log  0.00648 loglog  7.81170-10 

C,=    .log  0.99583 

C^=    log  1.00231 


104  EXTEEIOE  BALLISTICS 

Xow  we  see  that  0^  =  0^,  and  that  we  have  therefore  reached  the  limit  of  accuracy 
possible  by  the  method  of  successive  approximations  and  have  verified  the  value  of  C, 
and  no  further  work  in  this  connection  is  therefore  necessary.  We  therefore  have  for 
further  work  in  connection  with  this  trajectory : 

<^  =  4°   13'  14"         Z  =  2984.1         log  (7=1.00231 

To  determine  the  angle  of  fall,  time  of  flight  and  striking  velocity,  we  have, 
from  (116),  (117)  and  (118) 

tan  u)  =  B'  tan  0         T  =  CT'  sec  (f>         Vu  =  Uu  cos  </>  sec  w 

From  Table  II 

log5'  =  .10371         r  =  1.2333         w^  =  2026.1 

B'=    log  0.10371 

C=    log  1.00231 

<^  =  4°   13'  14"    tan  8.86803-10.  .sec  0.00118 cos  9.99882-10 

r'  =  1.2332    log  0.09103 

w^  =  2026.1    log  3.30666 


,  =  5°  21'  11" tan  8.97174-10 sec  0.00190 


r  =  12.43    log  1.09452 

ra,  =  2029    log  3.30738 

Hence  we  have  as  the  solutions  to  this  problem 

«/)  =  4°   13'  14". 
a>  =  5°   21'  11". 
T=  12.43  seconds. 
i;a;  =  2029  f.  s. 

These  are  the  correct  and  final  results,  and  it  will  be  observed  that  they  are  the 
values  which  appear  in  Columns  2,  3,  4  and  5  of  the  range  table  for  this  gun,  for  a 
range  of  10,000  yards.  We  have  therefore  learned  how  to  compute  the  values  for  these 
columns  in  the  range  tables.  In  a  later  chapter  will  be  given  the  forms  used  by  the 
computers  in  actually  computing  the  data  for  the  range  tables. 

Note. — The  mathematical  processes  carried  througli  in  the  preceding  cliapters  may  be 
briefly  and  generally  described  as  follows: 

1.  Considering  the  forces  acting  on  the  projectile  in  flight,  that  is,  the  force  of  gravity 
and  the  atmospheric  resistance,  and  dealing  with  differential  increments  at  any  point  of 
the  trajectory,  certain  equations  are  derived  (from  the  laws  of  physics  governing  motion) 
which  show  the  relations  existing  between  these  differential  increments  in  the  different 
elements  of  the  trajectory  at  the  given  point.  These  are  not  equations  to  the  curve  itself 
as  a  whole,  but  simply  express  the  relations  between  the  differential  increments  referred  to 
above.  Could  they  be  integrated  in  general  form,  they  could  be  generally  used  for  solu- 
tions, but  such  integration  is  impossible  owing  to  fractional  exponents,  and  some  other 
method  must  be  adopted.  The  accepted  method  is  known  as  Siacci's  method,  from  its 
deviser,  its  essential  point  being  the  introduction  into  the  computations  of  a  new  quantity 
known  as  the  "  pseudo  velocity,"  which  is  defined  by  saying  that  "  the  pseudo  velocity  at 
any  point  of  the  trajectory  is  the  component  of  the  remaining  velocity  at  that  point  in  a 
direction  parallel  to  the  original  line  of  projection."  By  the  introduction  of  this  quantity, 
it  becomes  possible  to  reduce  the  differential  equations  to  certain  others  that  are  known  as 
the  "  ballistic  formulfe."  which  are  used  in  the  practical  solutions  of  ballistic  problems. 
Each  of  these  formulae  contains  certain  integral  expressions,  which  are  represented  in  the 
formulae  by  the  symbols  A,  I,  S  and  T  (the  altitude,  inclination,  space  and  time  func- 
tions), and  the  values  of  these  functions  for  any  given  velocity,  whether  real  or  pseudo, 
may  be  found  in  Table  I  of  the  Ballistic  Tables.     That  is,  the  tabulated  values  of  these 


PRACTICAL  METHODS  105 

functions  are  simply  the  values  of  the  given  integral  expressions  when  integrated  between 
the  proper  limits  for  the  given  velocity. 

2.  As  stated  above,  these  several  functions  are  merely  values  of  certain  rather  involved 
integral  expressions,  the  values  of  which  for  any  given  velocity  may  be  found  in  Table  I. 
Different  subscripts  to  the  symbols  are  used  to  represent  the  values  at  different  points  of 
the  trajectory;  thus  -S'^^is  the  value  of  the  space  function  for  the  point  of  fall,  So  for  the 
vertex,  etc. 

3.  Professor  Alger  took  the  ballistic  formulae  as  they  stood  after  the  reductions 
described  above,  and  put  them  in  the  form  desired  for  any  particular  problem.  Any  neces- 
sary changes  for  this  purpose  were  simply  algebraic  and  trigonometrical  transformations  in 
order  to  get  the  value  of  the  desired  unknown  quantity  expressed  in  terms  of  the  ones  that 
are  known.  He  then  solved  in  each  case,  taking  the  necessary  values  of  the  integral 
expressions  I,  A,  S  and  T  from  Table  I. 

4.  The  successive  approximation  feature  (and  this  description  fits  every  such  case, 
whether  Alger  or  Ingalls)  becomes  necessary  because,  when  we  start  a  ballistic  problem 
we  usually  do  not  know  the  maximum  height  of  flight  of  the  projectile.  We  therefore  work 
the  problem  by  first  disregarding  this  height,  that  is,  by  considering  that  the  density  of 
the  air  throughout  the  flight  is  constant  and  equal  to  that  at  the  gun.  We  know  this  to  be 
wrong,  however,  and  that  our  first  result  can  therefore  be  only  approximate.  Therefore  by 
using  our  first  result  we  determine  the  maximum  ordinate  (Alger  by  one  formula  and  the 
use  of  a  table;  and  Ingalls  by  the  use  of  different  formulae  but  by  computation  without 
the  use  of  any  table)  and  from  that  the  altitude  factor  of  the  ballistic  coefficient.  This 
approximate  value  of  /  we  apply  to  our  first  value  of  the  ballistic  coefficient,  and  then 
repeat  our  computations,  thereby  getting  a  second  result,  which  is  still  approximate,  but 
more  nearly  correct  than  the  first  one.  By  continually  repeating  this  process  we  will 
finally  get  to  a  point  where  repeated  computations  make  no  change  in  the  value  of  the  bal- 
listic coefficient,  and  at  that  point  the  limit  of  accuracy  of  our  methods,  whatever  they 
may  be,  has  been  reached,  and  our  result  is  as  nearly  correct  as  it  is  possible  to  get  by  the 
adopted  methods.  Taking  the  final  value  of  the  ballistic  coefficient  thus  obtained  as  cor- 
rect, we  can  then  proceed  to  the  final  solution  of  the  problem. 

5.  Ingalls  further  simplified  the  ballistic  formulae  so  that  their  use  would  be  less 
difficult.  In  these  formulae  there  are  certain  integral  expressions  involving  the  values  of 
7,  A,  S  and  T  for  the  point  of  fall  and  for  the  vertex,  that  is,  I^^,  !„,  etc.,  and  certain  con- 
stantly repeating  combinations  of  these  integrals.  Ingalls  substituted  for  these  con- 
stantly repeating  combinations  of  integrals  certain  other  quantities  which  he  called 
"  secondary  functions,"  and  represented  by  the  symbols  A,  A',  A",  B,  B',  T',  etc.,  and 
thereby  derived  new  and  simpler  formulae  involving  those  secondary  functions  and  the 
pseudo  velocity,  u.  He  also  computed  the  values  of  the  secondary  functions  with  the 
expressions  integrated  between  all  useful  limits  (that  is,  of  the  integral  forms  which  they 
represent)  and  tabulated  them  in  Table  II  of  the  Ballistic  Tables.  To  solve  by  his 
methods,  we  therefore  take  his  forms  of  the  ballistic  formulae,  properly  transposed  to  put 
all  known  quantities  in  the  right-hand  member  and  the  desired  unknown  quantity  as  the 
left-hand  member  of  each  expression,  and  then  solve;  taking  the  values  of  the  secondary 
functions  that  we  need  from  Table  II.  To  use  this  table  we  must  know  for  use  as  argu- 
ments the  value  of  the  initial  velocity  and  also  the  quotient  of  the  horizontal  distance 

traveled  divided  by  the  ballistic  coefficient,  that  is,  ot  z  =  -^-  or  Z  =-^ 

The  above  paragraphs  of  this  note  indicate  the  manner  in  which  a  student 
shouhl  try  to  retain  in  his  mind  the  general  features  of  the  mathematical  processes 
described  in  this  text  book.  Each  student,  in  addition  to  learning  the  text  of  each 
chapter  in  detail  should  endeavor  to  formulate  in  his  own  mind  a  general  under- 
standing of  the  processes  described  in  the  chapter  in  accordance  with  the  general 
method  illustrated  above  for  the  fundamental  processes  of  exterior  ballistics. 


106 


EXTERIOE  BALLISTICS 


EXAMPLES. 

1.  Given  the  values  of  V  and  Z  contained  in  the  first  two  columns  of  the  follow- 
ing table,  take  from  Table  II  of  the  Ballistic  Tables  the  corresponding  values  of 
A,  A',  A",  B,  log  B',  u,  r  and  D' . 


DATA. 

ANSWERS. 

Problem. 

^=4- 

V. 

A.    A'. 

A". 

B. 

log  5'. 

u. 

7". 

D'. 

1 

3370 

1150 

.09855 

.21279 

1809.9 

.11431 

.06440 

914.07 

3.3323 

192.10 

2 

1763 

1150 

.04755 

.09963 

922.02 

.0.5214 

.03935 

1002.5 

1.6510 

46.780 

3 

6982 

11-50 

.23947 

.54002 

.30961 

.11151 

763.85 

7.6714 

1055.3 

4 

1326 

2000 

.01200 

.02550 

700.56 

.01,354 

.04989 

1683.3 

0.72.534 

9.2600 

5 

4173 

2000 

.04992 

.12123 

2454.9 

.07132 

.15484 

1175.2 

2.7711 

136.57 

6 

7652 

2000 

.12736 

.33625 

4753.3 

.20885 

.21477 

919.40 

0.1727 

786.12 

7 

1943 

2000 

.01087 

.02365 

1047.9 

.01274 

.06869 

20.50.8 

0.84207 

11.860 

8 

3756 

2600 

.02489 

.05904 

2173.0 

.0.3414 

.13693 

1619.7 

1.8427 

59.240 

9 

9743 

2600 

.12187 

.35927 

6438.1 

.23739 

.28950 

921.85 

7.0604 

1139.1 

10 

10742 

2600 

.14759 

.43913 

7131.0 

.29149 

.29561 

873.90 

8.1739 

1599.4 

11 

1818 

2700 

.00931 

.02005 

974.44 

.01074 

.06345 

2169.1 

0.7.5228 

9.1800 

12 

4747 

2700 

.03210 

.07981 

2840.9 

.04774 

.17323 

1484.6 

2.3925 

101.82 

13 

5561 

2700 

.04095 

.10630 

3421.5 

.06536 

.20376 

13.32.6 

2.9718 

162.49 

14 

7937 

2700 

.07613 

.21928 

5183.0 

. 14322 

.274.50 

1050.0 

.5.0092 

514.14 

15 

9541 

2700 

. 10868 

.32305 

6336.1 

.21501 

.29617 

948.95 

6.6211 

988.58 

16 

1856 

2900 

.00823 

.01777 

994.92 

.009.59 

.06326 

2331.8 

0.71408 

8.5600 

17 

2942 

2900 

.01434 

.03253 

1643.0 

.01818 

.10215 

2037.1 

1.2126 

24.840 

18 

3839 

2900 

.02034 

.04814 

2216.7 

.02787 

.13558 

1813.6 

1.6795 

.50.170 

19 

4815 

2900 

.02809 

.06988 

2879.5 

.04176 

.17277 

1595.8 

2.2.535 

90.750 

20 

8634 

2900 

.07677 

.22861 

5735.5 

.15189 

.29651 

1044.3 

5.. 301 6 

592.50 

21 

3231 

3100 

.01404 

.03213 

1819.5 

.01808 

.10982 

2119.6 

1.2626 

27.620 

22 

5742 

3100 

.03182 

.08283 

3534.7 

.05106 

.20430 

1.5.30.2 

2.6633 

129.36 

23 

8841 

3100 

.06943 

.21036 

.5923.6 

.14094 

.30760 

1076.3 

5.1291 

553.84 

24 

10305 

3100 

.09566 

.30074 

7027.7 

.20504 

.33126 

974.70 

6.5632 

998.90 

Note  for  Instructor. — In  exercising  class  in  these  interpolations,  give  to  each  mid- 
shipman one  problem  from  each  of  the  tables  given  in  tliis  and  in  the  following  five 
examples. 

2.  Given  the  values  of  V  and  Z  contained  in  the  two  first  columns  of  the  follow- 
ing table,  take  from  Table  II  of  the  Ballistic  Tables  the  corresponding  values  of 
A,  A',  A",  B,  log  B',  u,  r  and  D' . 


DATA. 

ANSWERS. 

Problem. 

-I- 

V. 

A. 

A'. 

A". 

B. 

log  B'. 

u. 

T'. 

D'. 

1 

2200 

1162 

.05974 

.12640 

1160.4 

.00666 

.04798 

981.92 

2.0769 

73.800 

2 

5500 

1173 

.  172.57 

..38758 

3051.4 

.21499 

.09544 

826.98 

5.7299 

.581.96 

3 

8100 

1187 

.28019 

.65643 

.37621 

.12808 

735.41 

9.0164 

1492.5 

4.. 

1800 

2030 

.016.54 

.03592 

970.00 

.01936 

.06810 

1606.6 

0.99900 

17.400 

5.  . 

4200 
7700 

2057 
2082 

.04764 
. 12004 

.11600 
..32104 

2475.3 
4821.9 

.06828 
.20103 

.15644 
. 22389 

1200.5 
932.58 

2.7176 
6.0.389 

131.02 

6 

755 . 90 

7 

3100 

2618 

.01902 

.04359 

1749.0 

.02456 

.11167 

1779.7 

1.4399 

35.460 

8.. 

7.300 

2643 

.068.32 

.19222 

4702.7 

.  12.389 

.25853 

1087.5 

4.5167 

403.81 

9 

9400 

2663 

.108.54 

.32104 

6222 . 3 

.21254 

.29194 

950.71 

6.5537 

961.42 

10 

5100 

2730 

.03494 

.08853 

.3090.0 

.05364 

.18625 

14.33.0 

2.6048 

122.00 

11. 

6100 

2750 

.04.581 

.12240 

3818.0 

.07660 

.22305 

1271.0 

3.3235 

205.50 

12 

8700 

2779 

.08.543 

. 25263 

57.59.1 

.16723 

.29158 

1012.8 

5.5986 

668.13 

13 

5500 

2913 

.03410 

.08787 

3368.0 

.05378 

.19836 

1465.8 

2 . 6889 

131.. 57 

14 

8200 

2954 

.06667 

.19629 

5415.4 

.12959 

.28890 

1093.6 

4.7975 

471.02 

15 

9700 

2982 

.09156 

.28154 

6545.6 

.19002 

.31698 

989.70 

6.1909 

860.34 

16 

4600 

3140 

.02210 

.0,5408 

2722.0 

.03198 

.16032 

1804.6 

1.9410 

66.000 

17 

7000 

3150 

.04291 

.119.30 

4486.0 

.07645 

.25060 

1322.5 

3.4925 

2.32.00 

18 

9900 

3160 

.08423 

.26398 

6741.8 

.17978 

.32934 

1009.4 

6.0338 

819.60 

PEACTICAL  METHODS 


107 


3.  Given  the  values  of  V  and  Z  contained  in  the  two  first  columns  of  the  follow- 
ing table,  take  from  Table  II  of  the  Ballistic  Tables  the  corresponding  values  of 
A°A',  A",  B,  log  B',  u,  r  and  B' . 


DATA. 

ANSWERS. 

Problem. 

.=4. 

F. 

A. 

A'. 

A". 

B. 

logB'. 

u. 

r. 

D'. 

1 

2730 

1157 

.07670 

.16392 

1452.6 

.08716 

.05583 

9.50.07 

2.0336 

119.16 

o 

5980 

1169 

. 19258 

.43558 

3336.81.24299 

.10109 

806.72 

6.3303 

713.94 

3 

8730 

1182 

.31173 

.73709 

.42530 

.13503 

712.68 

9.9097 

1814.7 

4 

1936 

2028 

.01807 

.03947 

id49.2 

.02138 

.07.330 

1576.1 

1.0858 

20.100 

5 

4757 

2048 

.05773 

.14368 

2846.4 

.08599 

.17296 

1131.5 

3.2098 

187.20 

6 

7915 

2063 

.12780 

.34164 

4955.5 

.21386 

.22372 

918.22 

6.3151 

833.56 

7 

3342 

2628 

.02082 

.04836 

1903.9 

.02753 

.12076 

1731.1 

1.5724 

42.840 

8 

7539 

2644 

.07243 

.20534 

4879.8 

.13292 

.26385 

1067.8 

4.7373 

450.35 

9 

9526 

2684 

.10970 

.32588 

6319.0 

.21624 

.29471 

946.98 

6.6413 

994.18 

10 

5433 

2733 

.03848 

.09921 

3328.8 

.06078 

.19856 

1373.2 

2.8391 

147.01 

11 

6214 

2748 

.04734 

.12721 

3902 . 4 

.07993 

.22720 

1253.1 

3.4172 

218.04 

12 

8848 

2763 

.08944 

.26494 

5861.8 

.  175.52 

29272 

1000.7 

5.7785 

719.75 

13 

5584 

2925 

.03464 

.08970 

3429.3  .05506|. 20142 

1457.0 

2.7347 

136.13 

14 

8282 

2944 

.068.53 

.20229 

.5475. 9;.  13369 

.29045 

1084.3 

4.8913 

492.76 

15 

9748 

2962 

.09391 

.28840 

6574. 2 '.194.52 

.31627 

983.28 

6.2813 

890.22 

16 

4632 

3148 

.02221 

.05445 

,2744.1 .03224 

.16136 

1802.8 

1.9536 

67 . 200 

17 

7148 

3155 

.04430 

.12447 

'4601.4  .08010 

. 25606 

1300.2 

3.5998 

248.31 

18 

9923 

3163 

.08449 

.26497 

,6759.5  .18052 

.32979 

1008.6 

6.0511 

825.65 

4.  Given  the  values  of  V  contained  in  the  first  column  and  of  the  secondary  func- 
tions contained  in  the  second  column  of  the  following  table,  take  from  Table  II  of 
the  Ballistic  Tables  the  corresponding  values  of  Z. 


108 


EXTERIOR  BALLISTICS 


5.  Given  the  value  of  Z  contained  in  the  first  column,  the  value  of  the  secondary 
function  contained  in  the  second  column,  and  the  limits  near  which  the  vahie  of  Y 
lies  contained  in  the  third  column  of  the  following  table,  take  from  Table  II  of  the 
Ballistic  Tables  the  corresponding  value  of  Y . 


Problem. 


DATA. 


1 

1732 

2 

3 

4140 
5615 

4 

1232 

5 

43S1 

6 

8175 

7 

2222 

8 

4444 

9 

8888 

10 

2551 

11 

5743 

12 

9107 

13 

3232 

14 

6474 

15 

9876 

16 

13.34 

17 

4321 

18 

8448 

Secondary  function. 

A 

=        0 

04632 

A' 

=       0 

27837 

A" 

=  3127 

5 

B 

=       0 

01273 

u 

=  1154 

7 

T' 

=        6 

7324 

A 

=        0 

01278 

A' 

=        0 

07735 

A" 

=  5796 

7 

B 

=       0 

01693 

u 

=  1298 

4 

T' 

=        6 

2333 

A 

=        0 

01607 

A' 

=        0 

12087 

A" 

=  6593 

3 

B 

=       0 

00555 

u 

=  1835 

2 

r 

=       4 

7867 

Limits  of  Y.* 


1150- 
1150- 
1150- 
2000- 
2000- 
2000- 
2600- 
2600- 
2600- 
2700- 
2700- 
2700- 
2900- 
2900- 
2900- 
3100- 
3100- 
3100- 


-1200 
-1200 
-1200 
•2100 
-2100 
■2100 
■2700 
■2700 
■2700 
•2800 
■2800 
■2800 
-3000 
-3000 
-3000 
-3200 
-3200 
-3200 


ANSWEKS. 


1154.9 
1137.0 
1195.6 
1962.4 
2007.4 
2007.5 
2595 . 9 
2602.0 
2469.1 
2715.0 
2693.0 
2668.9 
2909.6 
2016.8 
2727 . 6 
3080.0 
3089.0 
3090.1 


*  These  limits  determine  the  table  to  be  used;  In  some  cases  it  will  be  found  that  tlie 
interpolation  gives  a  value  of  Y  lying  outside  of  the  limits  indicated. 


6.  Given  the  values  of  Y  and  of  A'  contained  in  the  two  first  columns  of  the 
following  table,  take  from  Table  II  of  the  Ballistic  Tables  the  corresponding  values  of 
A",  without  determining  the  corresponding  value  of  Z. 


Problem. 

DATA. 

AXSWERS. 

Y. 

A'. 

A". 

1 

1150 
1179 
1192 
2000 
2053 
2086 
2600 
2677 
2689 
2700 
2750 
2772 
2900 
2932 
2988 
3118 
3150 
3173 

0.19787 
0.32995 
0.40843 
0.04932 
0.15563 
0.37.543 
0.05837 
0.12647 
0.27563 
0.02543 
0.10023 
0.00995 
0.03613 
0.13333 
0.30057 
0.20475 
0.27777 
0.02975 

1699.6 

9 

2692.0 

3 

3234.5 

4 

1236.5 

5 

3009.1 

6 

5293.2 

7 

21.54.4 

s 

37.52 . 0 

9 

5805 . 7 

10 

1194.3 

11 

3376.6 

1^ 

547.19 

13 

1786.5 

14 

4357.5 

15 

6760.3 

16 

5880.7 

17 

6877.5 

18 

1774.1 

PEACTICAL  METHODS 


109 


7.  Compute  by  Ingalls'  method  for  standard  atmospheric  conditions,  using 
successive  approximations,  the  values  of  the  angle  of  departure,  angle  of  fall,  time  of 
flight  and  striking  velocity  in  the  following  cases. 


DATA. 

ANSWERS. 

Problem. 

Projectile. 

Velocity. 

Range. 

0. 

T. 

V(j3. 

d 

w . 

f.  s. 

Yds. 

w 

Sees. 

f.s. 

In. 

Lbs. 

c. 

A 

3 

13 

1.00 

1150 

2130 

5° 

39.1' 

6° 

46' 

6.56 

867 

B 

3 

13 

1.00 

2700 

3720 

2 

59 . 3 

5 

33 

6.94 

1074 

C 

4 

33 

0.67 

2900 

3825 

1 

4.S.4 

2 

20 

4.98 

1843 

D 

0 

50 

1.00 

3150 

4370 

2 

11.4 

3 

44 

6.33 

1408 

E 

5 

50 

0.61 

3150 

4465 

1 

44.8 

2 

25 

5.44 

1941 

F 

6 

105 

0.61 

2600 

12690 

11 

03.4 

19 

38 

24.70 

1104 

G 

6 

105 

1.00 

2800 

3875 

1 

00.  / 

2 

41 

5.34 

1712 

H 

6 

105 

0.61 

2800 

3622 

1 

32.3 

1 

51 

4.46 

2134 

I 

7 

165 

1.00 

2700 

7230 

5 

00.7 

8 

31 

12.32 

1221 

J 

7 

165 

0.61 

2700 

7357 

3 

57.4 

5 

30 

10.54 

16.50 

K 

8 

260 

0.61 

2750 

8390 

4 

15.3 

5 

49 

11.62 

1735 

L 

10 

510 

1.00 

2700 

10310 

6 

49.8 

11 

09 

17.05 

1293 

M 

10 

510 

0.61 

2700 

11333 

6 

07.6 

8 

30 

16.30 

1653 

N 

12 

870 

0.61 

2900 

21650 

12 

30.9 

19 

55 

33.59 

1441 

0 

13 

1130 

1.00 

2000 

10370 

11 

15.2 

16 

09 

21.45 

1168 

P 

13 

1130 

0.74 

2000 

11111 

10 

58.0 

14 

44 

21.57 

1281 

Q 

14 

1400 

0.70 

2000 

14220 

14 

48.8 

20 

15 

28.68 

1251 

R 

14 

1400 

0.70 

2600 

14370 

8 

32.4 

11 

55 

21.76 

1577 

8.  Given  the  data  contained  in  the  first  eight  columns  of  the  following  table, 
compute  in  each  case  the  values  of  <^,  w,  T  and  v^^,  by  Ingalls'  method,  using  Table  II, 
and  using  in  each  case  the  value  of  /  from  Table  V  corresponding  to  the  maximum 
ordinate  given  in  the  table  below. 


DATA. 

ANSWERS. 

Problem. 

Projectile. 

Atmosphere. 

f.s. 

R. 

Yds. 

Max. 

T. 

Sees. 

d. 

10. 

Bar. 

Ther. 

ord. 
leet. 

0.  . 

w. 

f.s. 

In. 

Lbs. 

c. 

In. 

°F. 

A 

3 

13 

1.00 

Standard 

11.50 

2550 

265 

7° 

03.4' 

8° 

40' 

8.07 

832 

B 

3 

13 

1.00 

Standard 

2700 

3450 

158 

2 

36.4 

4 

42 

6.20 

1122 

C 

4 

33 

0.67 

Standard 

2900 

4000 

112 

1 

49.8 

2 

31 

5.27 

1802 

D 

5 

50 

1.00 

29.00 

20 

3150 

3870 

108 

1 

52.1 

3 

06 

5.46 

1475 

E 

5 

'50 

0.61 

29.50 

22 

31.50 

3850 

80 

1 

28.0 

1 

58 

4.61 

2011 

F 

6 

105 

0.61 

30.00 

25 

2600 

14530 

3798 

15 

42  .,4 

28 

31 

32.04 

1036 

G 

6 

105 

1.00 

30.1^ 

-  27 

2800 

4570 

169 

2 

33.0 

3 

55 

6.82 

1476 

H 

6 

105 

0.01 

30.25 

30 

2800 

4030 

101 

1 

47.2 

2 

14 

5.11 

2010 

1 

7 

165 

1.00 

.30.. 33 

33 

2700 

6030 

363 

3 

,55.2 

6 

22 

9.85 

1301 

J 

165 

0.61 

30.50 

35 

2700 

6540 

328 

3 

28.3 

4 

46 

9.29 

1676 

K 

8 

260 

0.61 

30.67 

40 

2750 

8080 

485 

4 

10.0 

5 

45 

11.32 

1698 

L 

10 

510 

1.00 

31.00 

45 

2700 

9090 

807 

5 

.52.9 

9 

28 

14.80 

1.320 

LI 

10 

510 

0.61 

30.75 

50 

2700 

10070 

784 

5 

20.3 

7 

17 

14.28 

1691 

N 

12 
13 

870 

0.61 

30.33 
30.25 

60 
70 

2900 

2000 

22030 
10560 

4801 
1937 

13 
11 

23.2 
39.1 

21 
16 

56 
51 

35.37 

22.08 

1377 

0 

1130 

1.00 

1154 

P 

13 

1130 

0.74 

29.50 

80 

2000 

11050 

1830 

10 

45.6 

14 

19 

21.24 

1299 

Q 

14 

1400 

0.70 

29.00 

90 

200C 

14020 

3204 

14 

09.7 

19 

00 

27.67 

1284 

R 

14 

1400 

0.70 

28.75 

100 

2600 

14590 

1960 

8 

23.2 

11 

23 

21.63 

1645 

110 


EXTERIOE  BALLISTICS 


9.  Compute  by  Alger's  method,  without  using  Table  II,  the  values  of  4>,  w,  T  and 
Vu,  from  the  data  contained  in  the  following  table,  correcting  for  altitude  in  each  case 
by  successive  approximations. 


DATA. 

ANSWERS. 

Prob- 

Projectile 

Atmosphere. 

Wind* 

lem. 

V. 

f.s. 

R'nge. 
Yds. 

T. 

Sees. 

f.s. 

d. 

w. 

Bar. 

Ther. 

Value 

pon't. 
f.s. 

0. 

w. 

In. 

Lbs 

c. 

In. 

°F. 

of  8. 

1 

3 

13 

Standard 

2800 

2000 

None 

1° 

00.9' 

1° 

25' 

2.79 

1671 

2 

5 

50 

Standard 

2550 

3000 

None 

1 

54.4 

2 

49 

4.76 

1439 

3 

6 

100 

Standard 

2300 

4000 

None 

3 

07.5 

4 

31 

6.97  1321 

4 

8 

250 

Standard 

2300 

4000 

None  i 

2 

45.1 

3 

36 

6.38 

1552 

5 

12 

850 

Standard 

2250 

5000 

None 

3 

26.0 

4 

15 

7.88 

1626 

6 

11.024 

760.4 

1.0306 

1733 

2260 

+  19| 

2 

21.5 

2 

36 

4.27 

1480 

7 

11.024 

760.4 

1.0058 

1733 

6788 

—  14 

8 

25.7 

11 

01 

14.64 

1173 

8 

6 

100 

30.05 

70 

2900 

9700 

None 

8 

53.3 

17 

58 

19.80 

987 

9 

12 

850 

30.19 

59 

2S27 

11566 

None  i 

6 

55.0 

11 

14 

18.16  1364 

10 

11.024 

760.4 

1.0174 

1733 

11207 

—  12; 

17 

32.7 

24 

33 

28.06  1046 

11 

15.75 

2028 

Standard 

1805 

1094 

None 

0 

57.8 

1 

00 

1.86  1712 

12 

15.75 

2028 

Standard 

1805 

3281 

None 

3 

06.2 

3 

27 

5.91  1542 

13 

15.75 

2028 

Standard 

1805 

5468 

None 

0 

36.4 

6 

42 

10.43  1391 

14 

3 

13 

Standard 

2800 

2000 

None 

1 

00.7 

1 

26 

2.79  1672 

15 

3 

15 

Standard 

2628 

1883 

None 

1 

00.0 

1 

21 

2.66  1720 

16 

5 

60 

Standard 

2900 

3000 

None 

1 

21.2 

1 

50 

3.91  1837 

17 

5 

55 

Standard 

■■ 

2997 

3095 

None 

1 

21.7 

1 

00 

4.02  1796 

*  The  sign  +  means  a  wind  with  the  flight  of  the  projectile,  and  a  —  sign  a  wind 
against  it.  Therefore,  in  problem  6,  say,  in  order  to  get  the  desired  range  we  would  have 
to  proceed  as  though  the  initial  velocity  were  really  1733  — 19  =  1714  f.  s.  and  there  were 
no  wind,  and  compute  results  accordingly. 

Note. — The  above  problems  in  Example  9  are  taken  from  Alger's  text  book,  and  cover 
guns  of  older  date,  both  U.  S.  Navy  and  foreign.  Note  the  difference  between  this  data  and 
modern  weights  and  velocities;  and  observe  care  to  use  correct  data  as  given  in  the  table. 


X 

=  tan<f,      ,    ^,     li''~ 

2  COS-^  \bu  — 

M 

tan^: 

=  tan0— ;rT   {lu- 

2  COS-  </> 

-Iv) 

and 

by  el 

iminatiiig  Iv  from  the  above  we  get 

a;                    2  cos-  <j>  \ 

-  — 

-Ay 

-Sv. 

159. 

Equation  (76) 

is 

CHAPTEE  9. 

THE  DERIVATION  AND  USE  OF  SPECIAL  FORMTJL^  FOR  FINDING  THE 
COORDINATES  OF  THE  VERTEX  AND  THE  TIME  OF  FLIGHT  TO  AND 
THE  REMAINING  VELOCITY  AT  THE  VERTEX,  FOR  A  GIVEN  ANGLE 
OF  DEPARTURE  AND  INITIAL  VELOCITY,  WHICH  INCLUDES  THE  DATA 
GIVEN  IN  COLUMN  8  OF  THE  RANGE  TABLES. 

158.  Equations  (74)  and  (75)  are  BaUistie 

^  ^       '  ^       '  formula. 

(132) 
(133) 

(134) 


t  =  C  sec  <p{Tu-Ty)  (135) 

160.  Equations  (132),  (133),  (134)  and  (135)  may  be  written 

-^=tanc/)--^^  =tan^+— ^  (136) 

X  2cos-(^  2  cos- </> 

tan^  =  tan(^-— ^^  (137) 

2  cos-  cf> 

t  =  Ct' sec  (f>  (138) 

by  the  introduction  of  the  general  forms,  a,  a',  h  and  /'  of  Ingalls'  secondary  func- 
tions, as  explained  in  the  last  chapter. 

161.  Equations  (136)  and  (137)  may  be  written  Transforma- 

tion of  equa- 

l=ta„^(l--^)  (139)    """■ 

X  ^  \        sm  2(f)/ 

tan^  =  tan<^fl--A?-')  (140) 

\         sm  26J 

Substituting  in  these  the  value  of  sin  2^  =  .4C'  from  (115),  we  get 

l=^(.4-a)  =  — ^(.4-a)  (141) 

X  A     ^  ^2  cos'  (^  ^  ^  ^        ^ 

tane=^-^^  (A-a')  =  —^  (A -a')  (142) 

A  2  COS"  <^ 

162.  Xow  by  taking  the  iirst  two  members  of  each  of  the  above  equations,  that  is,   Equations  for 

ordinate  and 
i.„„    I  inclination, 

y=^^^iA-a)x  (143) 

tan^=  ^^  (A-a')  (144) 

we  can  readily  find  the  values  of  y  and  9  corresponding  to  any  given  value  of  x  for 
any  given  trajectory;  that  is,  by  computing  the  ordinates  and  angles  of  inclination 
corresponding  to  any  necessary  number  of  abscissse,  we  are  in  a  position  to  actually 
plot  the  trajectory  to  scale,  provided  we  have  determined  or  know  the  values  of  <;!>, 


113  EXTERIOR  BALLISTICS 

V,  X  and  C  for  that  trajectory;  for  a  knowledge  of  the  values  of  V  and  oi  Z=  -^ 
is  necessary  to  enable  us  to  use  Table  II. 

163.  The  quantity  a  varies  with  x,  and  must  be  taken  from  the  "  A  "  column  of 

Table  II  with  V  and  2=  yp  as  arguments.     Similarly,  a'  must  be  taken  from  the 

"  A' "  column  with  the  same  arguments. 

164.  Eor  the  vertex,  we  know  that  6  =  0,  and  (14^)  therefore  becomes,  for  that 
particular  point, 

i^(l-ao')=0 
A. 

and,  as  — ~  cannot  be  equal  to  zero,  then  we  must  have 

A-ffo'  =  0     or     A=ao'  (145) 

165.  Also,  if  we  suppose  6=  —(j>  at  some  point  in  the  descending  branch  of  the 
trajectory,  which  point  manifestly  exists,  as  in  that  branch  the  value  of  6  varies  from 
zero  at  the  vertex  to  —  w  at  the  point  of  fall,  and  we  have  seen  that  w  is  always 
numerically  greater  than  <f>,  equation  (142)  will  become  for  that  point 

tan(-<^)  =  ^(4_^'_^) 

or  A—a'_^=—A 

or  a'_^  =  2A  (146) 

whence,  from  (145)  and  (146) 

ao=A=U'_^ 

166.  Substituting  a^'  for  A  in  (141)  and  designating  symbols  relating  to  the 
vertex  by  the  subscript  zero,  we  get 

-%  =  ^^^^0  tan  <j>=^tan<l> 

Xq  ttQ  CIq 

whence  2/o  =  -^  tan  cf>  =  C  -^  tan  </> 

The  secondary  167.  Now  if  we  let  A"  =  ^  =  ^%  in  which  A"  may  be  taken  from  Table  II, 

function  A"  tt^L  tt^ 

using  V  and  ao'  =  A=  - — j~  ,  we  will  have  the  expression  for  the  summit  ordinate, 

or  ordinate  of  the  vertex 

i/,  =  r  =  A"C'tan<^  (147) 

It  will  be  observed  that  a^'  is  a  special  value  of  A',  this  latter  symbol  referring  to  the 
entire  trajectory.  This  value  of  the  ordinate  at  the  vertex,  y^,  is  ordinarily  denoted 
by  Y  in  work. 

168.  We  have  already  shown  that,  for  the  whole  trajectory,  A  =  - — ^y     >  ^^<i  ^^so 

that,  for  the  vertex  of  the  curve,  a^'  =  A,  and  from  this  we  see  that,  for  that  'particular 

point  aQ=  — yy^  ;  and  we  also  know  that  Qq  is  merely  the  special  value  of  A'  for  that 

particular  point  in  the  trajectory,  namely,  the  vertex.  Hence  if  we  know  the  values  of 
(f>,  V  and  C,  we  can  compute  this  particular  value  of  A',  namely,  ag,  from  the  ex- 
pression a(,'=  - — -^  ;  and  then,  as  this  is  a  special  value  of  A',  we  may  look  for  it  in 
the  A'  column  of  Table  II,  and  by  interpolation  in  the  usual  manner  we  may  take 


PRACTICAL  METHODS 


113 


from  that  table  the  corresponding  value  of  A" ;  and,  in  fact,  the  corresponding  values 

of  any  other  of  the  secondary  functions  for  the  vertex.     This  explains  the  reasons 

for  the  method  of  determining  the  value  of  A"  described  and  used  in  the  last  chapter.* 

169.  We  also  find,  in  the  Z  column  corresponding  to  the  above  interpolation. 


the  value  of  Zq  =  -yf ,  and  we  therefore  have 


,  =  Cz„ 


(148) 


170.  Assembling  the  equations  already  derived,  we  see  that  our  formulas  for   Equations 
>^  i  -^  '  for  vertex. 


finding  the  elements  at  the  vertex  are 

y,  =  Y  =  A"C  tan  cl> 


Xo  =  Czo 


t^^CtQ  sec  <i> 
Vq  =  Wq  cos  4> 


(149) 
(150) 
(151) 
(153) 


171.  Let  us  now  proceed  with  our  standard  problem,  the  12"  gun,  for  which 
"F  =  2900  f.  s.,  ?t'  =  S70  pounds  and  c  =  O.Gl,  for  which,  at  10,000  yards  range,  we 
have  already  determined  in  the  last  chapter  that  log  (7  =  1,00331,  Z=:2984.1  and 
<^  =  4°   13'  14". 

tQ=:Ct'  sec  (f>  Vo=:U(yCOS<f> 


Y  =  A"Ct&nc(> 
From  Table  II 


Xn  —  yyZn 


4  =  .01408+»^  =.014601 


4"  =  849- 


1.99x56 
13 


=  838.87 


'for  the  entire  trajectory,  which  equals  a^'  for 
the  vertex.    This  could  also  be  determined 

by  solving  A  =  — j~  for  the  above  values 

instead  of  taking  it  from  the  table. 

by  using  the  value  of  A'  given  above  as  an 
argument  in  the  A'  column,  and  working 
with  the  nearest  tabular  value.  (Ordi- 
narily work  from  the  next  lower  tabular 
value,  however.) 


500 _  ,000199X100^  J. gj^g 


io  =  .565  + 


Wn  =  2435- 


.0011 

.041x81.9 
100 

30x81.9 


=  .59858 


100 


=  3410.4 


C=    log  1.00231 log  1.00231.  .log  1.00231 

.1"  =  838.87   log  2.92370 

<^  =  4°13'14"    ...tan  8.86803-10 sec  0.00116 cos  9.99882-10 

Zo  =  1581.9    log  3.19918 

V  =  .59858    log  9.77712-10 

Mn  =  2410.4 loff  3.38209 


r=  622.36    log  2.79404 


a:o  =  15903.3    log  4.20149 

^0  =  6.034    log  0.78061         " 

fo  =  2403.9   log  3.38091 

a:o  =  5301.1  yards.  ^o  =  6.034  seconds. 

F=  622.36  feet.  fn  =  2403.9  foot-seconds. 


*  See  foot-note  to  paragraph  152. 
8 


114  EXTEEIOE  BALLISTICS 

172.  Had  we  not  known  the  correct  value  of  C\  as  corrected  for  altitude,  but 
had  only  known  <f}  and  V,  the  work  would  then  have  been  by  successive  approxima- 
tions, as  follows : 

F  =  2900f.  s.         <^  =  4°   13'  14"         w  =  870  pounds         c  =  0.61 
Y  =  A"Ctsin<j>         loglog /  =  log  F  + 5.01765 -10         x^  =  Cz^ 
tg  =  Ctf^'  sec  (f>        Vq  =  UqC0S(}) 

w  =  S70    log  2.93953 

c=0.61    log  9.78533-10.  .colog  0.21467 

d-  =  U4:    log  2.15836 colog  7.84164-10 

C,=    log  0.99583 colog  9.00417-10 

2</>  =  S°  26'  28"    sin  9.16670-10 

Go/  =  .01482    log  8.17087  - 10 

From  Table  II,  with  .01482  in  the  A'  column  as  an  argument, 

A,"  =  8id+  ^^^  X  ^  =849  +  1  =  850 

A,"  =  SoO    log  2.92942 

C\=     log  0.99503 

</)  =  4°   13'  14"    tan  8.86803-10 

Zi=    log  2.79328 

Constant    log  5.01765  - 10 

/i=     log  0.00647 loglog  7.81093-10 

Ci=    log  0.99583 

Co=     log  1.00230 colog  8.99770-10 

2(^  =  8°  26'  28"    sin  9.16670-10 

a.,/  =  .0146    log  8.16440  - 10 

From  Table  II  as  above 

A."  =  793+  ^^-^  =793  +  46  =  839 

A."  =  SSd    log  2.92376 

C.=    log  1.00230 

(6  =  4°   13'  14"    tan  8.86803-10 


72  =  622.43   log  2.79409 

Constant    log  5.01765  - 10 


/,=     log  0.00648 loglog  7.81174-10 

Ci=     log  0.99583 

(7,=    log  1.00231 colog  8.99769-10 

2</>  =  8°   26'  28"    sin  9.16670-10 

flo/  =  .0146    log  8.16439  - 10 

As  these  last  two  successive  values  of  a^  are  equal,  we  have  evidently  reached  the 
limit  of  accuracy  in  our  approximations,  and  we  have  for  the  remainder  of  the 
problem 

a '  =  .0146     and     log  C  =  1.00231 


PEACTICAL  METHODS  115 

Also,  from  log  Yn  as  found  above,  we  have  that  F  =  622.43  feet,  and  from  Table  II 
2o  =  1581.8         V  =  -59854         Mo  =  2410.5 

C=     log  1.00231 log  1.00231 

(fy  =  4°   13'  14"    sec  0.00118 cos  9.99882-10 

2o  =  15S1.8    log  3.19915 

V  =  . 59854    log  9.77709-10 

w„  =  2410.5    log  3.38093 


a-,  =  15902.3    log  4.20146 


L  =  6.0336    lose  0.78058 


t'o  =  2404.0    log  3.38093 

a;o  =  5300.7  yards. 
r  =  622.43  feet. 
^0  =  6.0366  seconds, 
i;^,^  2404.0  foot-seconds. 

173.  If  we  desire  to  plot  any  particular  trajectory  to  scale,  we  can  determine 
the  ordinate  corresponding  to  any  given  abscissa,  and  also  the  angle  of  inclination 
of  the  curve  at  the  given  point  as  follows : 

We  have  from  (143)  and  (144) 

y=  :t^  (A-a)x  (153) 

and  tan  e=  '^-^  (A-a')  '  (154) 

A 

Suppose  we  wish  to  plot  the  10,000-yard  trajectory  for  our  standard  problem,  for 
which  we  now  know  that  c/>  =  4°   13'  14"  and  log  (7  =  1.00231. 

2cj>  =  8°   26'  28"    sin  9.16670  - 10 

C=     colog  8.99769-10 

1  =  .014601    log  8.16439-10 

a^'  =  A  =  .01i601 
^.^,,3^.009^01^X56  ^g33_g,         ,,  =  1500+  ^^^  X^-^^^^^"^  =1581.9 

^"  =  838.87    log  2.92370 

C=    log  1.00231 log  1.00231 

<^  =  4°   13'  14"    tan  8.86803-10 

2„=  1581.9    log  3.19918 


y/„  =  F  =  622.35    log  2.79404 

a:o  =  15903    log  4.20149 

The  coordinates  of  the  vertex  are  therefore  5301  yards  in  range  and  622.35  feet 
in  altitude. 

174.  In   the   equations  given   we  now   find  the   value   of       /^  for  the  given 
trajectory,  having  the  value  of  A  as  above. 

<^  =  4°   13'  14"    log  8.86803-10 

A  =  .014601    log  8.16439-10 colog  1.83561 

t_anj.  ^5  Q5^    j^^  0.70364 

and  our  equations  become 

^  =  5.054(.014601-a)x     tan  ^  =  5.054(.014601-a') 


116 


EXTERIOE  BALLISTICS 


175.  The  following  table  gives  the  results  of  work  with  these  equations  for 
abscissa  varying  by  1000  yards  for  this  trajectory,  some  of  the  cases  being  worked 
out  below : 


Abscissae. 

Ordinates. 

Remarks. 

Yards. 

Feet. 

ti. 

0 

0 

4° 

13' 

14" 

Origin. 

1000 

203.58 

3 

31 

57 

2000 

370.06 

2 

48 

55 

3000 

497.25 

2 

02 

20 

4000 

,581.62 

1 

10 

37 

5000 

620.81 

0 

17 

11 

5301 

622.35 

0 

00 

00 

Vertex. 

6000 

610.34 

(— )0 

41 

35 

7000 

547.66 

(-)l 

43 

08 

8000 

427.21 

(— )2 

50 

43 

9000 

246.44 

(— )4 

04 

16 

10000 

0 

(-)5 

21 

34 

Point  of  fall;  6  =  - 

—  w. 

Work  for  3000  yards : 

C=    colog  3.99769-10 

.^  =  9000    los:  3.95^:34 


2  =  895.21    log  2.95193 

a  =  .00326+  -00043x95.21  ^Qogggg         ^'  =  .0067+  -0009x95.21   ^00^55^ 


A  =  .014601 
a  =  .003669 


100 
yl  =  .014601 

a'  =  .007557 


100 


4-a  =  .010932    log  8.03870-10 

A -a'  =  . 007044   ." lo^  7.84782-10 


tan  (^ 


A 


=  5.054 log  0.70364. 

:  =  9000 losr  3.95424 


.los:  0.70364 


2/  =  497.25   log  2.69658 

6  =  2°  02'  20"    tan  8.55146-10 

Work  for  8000  yards : 

C=    colog  8.99769-10 

a;  =  24000    log  4.38021 

2  =  2387.3    log  3.37790 

a=.01059+  •00056X87^_3  ^^^11079         a'  =  .0233+:^013x87J  ^^^^^^^g^ 


4  =  .014601 
o  =  .011079 


100 
^  = 

a'  = 


100 


.014601 
.024435 


4-a=.003522    log  7.54679-10 

A -a'=(-). 009834    (-)log  7.99273-10 


tan  <^ 


:5.054    log  0.70364, 


los:  0.70364 


a;  =  24000    log  4.38021 


i/  =  427.21    log  2.63064 

6={-)2°  50'  43"    (-)tan  8.69637-10 


PRACTICAL  METHODS 


117 


The  work  for  the  point  of  fall,  that  is,  for  an  abscissa  of  a:=  10,000  yards, 
becomes : 

C=     colog  8.99769-10 

a:  =  30000    log  4.47712 

2  =  2984.1    log  3.47481 

,^.01408+  m^l^  ..014601         a'  =  .0319+  ^^^^^^  =-033162 

A  =  .014601  A=         .014601 

a=. 014601  a'=         .033163 


4-a  =  0 

A_a'=(-).018561    (-)log  8.26860-10 

ifl^^=5  054  log  0.70364 

A  

^=(-)5°  21'  34"    (-)tan  8.97224-10 

y  =  0 

176.  A  reversal  of  the  original  formulge  ■would  enable  us  to  find  the  abscissa   Reverse 
corresponding  to  any  given  ordinate;  but  there  are  some  practical  difficulties  in  the 
way  of  a  simple  use  of  the  formulae  for  this  purpose,  and  as  it  is  not  a  usual  case  it  is 
not  considered  necessary  to  go  into  the  matter  here. 


EXAMPLES. 

1.  Given  the  data  contained  in  the  following  taljle,  compute  the  values  of 
To,  y^iY),  ^0  and  t^o  by  Ingalls'  method,  using  Table  II,  and  correcting  for  altitude 
in  each  case  by  computing  successive  approximations  to  the  value  of  C. 


DATA. 

ANSWERS. 

Problem. 

Projectile. 

Atmos 

phere. 

<p. 

Ve- 
locity, 
f.  s. 

Yds. 

2/o=r. 

Feet. 

Sees. 

Vq. 

d. 
In. 

w. 
Lbs. 

c. 

Bar. 
In. 

Tlier. 

f.  8. 

A 

3 

3 

4 

5 

5 

6 

6 

6 

7 

7 

8 

10 

10 

12 

13 

13 

14 

14 

13 

13 

33 

50 

50 

105 

105 

105 

165 

165 

260 

510 

510 

870 

1130 

1130 

1400 

1400 

1.00 
1.00 
0.67 
1.00 
0.61 
0.61 
1.00 
0.61 
1.00 
0.61 
0.61 
1.00 
0.61 
0.61 
1.00 
0.74 
0.70 
0.70 

28.00 
28.40 
29.15 
29.90 
30.00 
30.10 
30.70 
30.90 
31.00 
30.75 
30.50 
30.17 
30.00 
29.75 
29.33 
29.00 
28.50 
28.25 

100 
95 
93 
87 
84 
79 
75 
73 
67 
58 
49 
35 
29 
23 
18 
15 
10 
5 

6° 
2 
1 
1 
1 
14 
2 

1 

5 

3 

4 

6 

5 
13 
10 
10 
13 

8 

52' 

40 
50 
53 
45 
01 
18 
44 
20 
41 
03 
30 
52 
49 
40 
08 
54 
14 

54" 

24 

00 

12 

48 

06 

48 

06 

18 

36 

06 

06 

42 

48 

18 

42 

00 

06 

1150 
2700 
2900 
3150 
3150 
2600 
2800 
2800 
2700 
2700 
2750 
2700 
2700 
2900 
2000 
2000 
2000 
2600 

1310 
2106 
2196 
2272 
2453 
8529 
2407 
2091 
4227 
3747 
4292 
5483 
5849 
12661 
5353 
5542 
7245 
7468 

257 

171 

113 

127 

123 

3827 

160 

100 

686 

393 

493 

1067 

984 

5365 

1676 

IRIO 

2938 

1770 

3.88 
3.00 
2.55 
2.62 
2.65 

13.88 
3.00 
2.43 
6.01 
4.75 
5.31 
7.58 
7.49 

16.90 
9.65 
9.58 

12.89 

10.01 

958 

B 

1659 

c 

2300 

D 

21.54 

E. 

2460 

F 

1373 

G 

2075 

H 

2393 

I 

1667 

J 

2088 

K 

2154 

L 

M 

X 

0 

1766 
2054 
1803 
1418 

P 

1535 

Q 

1471 

R 

1950 

118 


EXTERIOR  BALLISTICS 


2.  Given  the  data  contaiiied  in  the  following  table,  compute  the  values  of 
Xo,  y^  {Y),  to  and  v^  by  Ingalls'  method,  using  Table  II,  and  correcting  for  /  for  the 
mean  altitude  during  flight  given  in  Column  8  of  the  table.  (Note  that  this  is  not 
the  maximum  ordinate.) 


DATA. 

ANSWERS. 

Prob- 
lem. 

P 

■ojectile. 

Atmosphere. 

<p. 

Ve- 
locity. 

Mean 

lieight 
of 

Yds. 

Feet. 

Sees. 

''"o- 

f.  S. 

d. 

if;. 

Bar. 

Ther. 

f.  s. 

flight. 

In. 

Lbs. 

In. 

°F. 

Feet. 

A 

3 

13 

1.00 

28.15 

0 

5° 

14' 

IS" 

1150 

100 

1032 

150 

2.97 

900 

B 

3 

13 

1.00 

28.75 

5 

9 

48 

54 

2700 

120 

2011 

170 

3.00 

1524 

C 

4 

33 

0.67 

29.00 

10 

1 

18 

42 

2900 

40 

1619 

59 

1.85 

2371 

D.... 

5 

50 

1.00 

29.30 

20 

1 

44 

12 

3150 

72 

2070 

106 

2.44 

2131 

E 

5 

50 

0.61 

29.70 

25 

1 

39 

24 

3150 

72 

2278 

107 

2.47 

2436 

F 

6 

105 

0.61 

29.90 

30 

9 

IS 

54 

2600 

1257 

6378 

1841 

9.77 

1514 

G 

6 

105 

1.00 

30.00 

40 

1 

48 

36 

2800 

68 

1971 

101 

2.41 

2167 

H.... 

6 

105 

0.61 

30.15 

50 

1 

37 

54 

2800 

60 

1976 

89 

2.29 

2416 

I 

7 

165 

1.00 

30.33 

60 

4 

38 

00 

2700 

360 

3856 

536 

5.34 

1744 

J  .... 

7 

165 

0.61 

30.50 

70 

3 

37 

18 

2700 

254 

3709 

381 

4.67 

2110 

K.... 

8 

260 

0.61 

30.67 

75 

4 

07 

12 

2750 

344 

4391 

512 

5.42 

2165 

L 

10 

510 

1.00 

30.90 

80 

6 

36 

24 

2700 

744 

5649 

1113 

7.76 

1789 

M.... 

10 

510 

0.61 

31.00 

85 

5 

57 

06 

2700 

680 

6009 

1019 

7.64 

2081 

N.... 

12 

870 

0.61 

30.75 

90 

9 

04 

36 

2900 

1706 

9655 

2564 

11. ()2 

2065 

0 

13 

1130 

1.00 

30.00 

95 

10 

21 

36 

2000 

1076 

5408 

1626 

9.56 

1471 

P 

13 

1130 

0.74 

29.50 

100 

10 

48 

54 

2000 

1220 

5998 

1851 

10.30 

1560 

Q 

14 

1400 

0.70 

29.00 

80 

14 

02 

42 

2000 

2018 

7472 

3043 

13.16 

1502 

E 

14 

1400 

0.70 

28.00 

60 

8 

23 

54 

2600 

1238 

7783 

1869 

10.33 

1989 

3.  Knowing  that  y='^-^  (A-a)x  and  that  tan  0=  ^r^  (A-a'),   derive 
A  A. 

expressions  for  the  values  of  y  and  tan  6  in  terms  of  a,  a',  x  and  of  the  numerical  coeffi- 
cients, for  any  point  in  each  of  the  trajectories  given  below,  atmospheric  conditions 
being  standard. 


Oh 


A.. 
E.. 
C. 
D.. 
E.. 
F.. 
G.. 
IT. 
I.. 
J.. 
K. 
L.. 
M. 
N. 
O.. 
P.. 
Q.. 
R.. 


DATA. 


Projectile. 


Lbs. 


13 

13 

33 

50 

50 

105 

105 

105 

165 

165 

260 

510 

510 

870 

11.30 

1130 

1400 


f.s. 


11.50 

2700 
2900 
3150 
3150 
2000 
2800 
2800 
2700 
2700 
2750 
2700 
2700 
2900 
2000 
2000 
2000 


n. 

Yds 


Value 

of 
loe  C. 


21,30  0 
37200 

3825  0 
43700 
4465j0 
126900 
38750 
3622|o 
72.300 
7357  0 


1400  0.7012600 


8390 
10310 
11333 
21650 
1037010 


11111 
14220 
14370 


.16152 
.16179 
.48936 
. 30272 
.5169? 
.70596 
.46608 
.68039 
.5.3374 
.74663 
.82912 
.71990 
.93343 
.04355 
.84469 
.97554 
.04355 
.02872 


39'  06" 
59  18 

43  24 
11  24 

44  48 
03  24 
55  48 
32  18 
00  42 

57  24 
15  18 
49  48 
07  36 
30  54 
15  12 

58  00 
48   48 

8    32  24 


ANSWERS. 


2/  = 


0. 73236 ( 
0. 72758  { 
1.54425 ( 
1. 00530 ( 
1. 64560 ( 
2. 63750 ( 
1. 46400 ( 
2. 39700  ( 
1.72170( 
2. 80300 ( 
3 . 39220 ( 
2.66130( 
4. 33860 ( 
5.79990  ( 
3 . 63520  ( 
4. 90370 ( 
5.91410( 
5. 46220 ( 


.135125- 
.071750- 
.019483- 
.038037- 
.018.531- 
.074085- 
.023017- 
.011203- 
.050924- 
.024675- 
.021932- 
.045010- 
.024739- 
.038272- 
.0.54735- 
.039516- 
.044718- 
.027492- 


-a)x 
■a)x 
-a)x 
■a)x 
-a)x 
■a)x 
-a)x 
-a)x 
-a)x 
-a)x 
■a)x 
-a)x 
-a)x 
-a)x 
•a)x 
-a)x 
-a)x 
-a)x 


tan  e  = 


0.73236( 
0.72758 ( 
1.. 54425 ( 
1. 00.530  ( 
1 . 64560 ( 
2 . 63750 ( 
1 . 46400 ( 
2. 39700 ( 
1.72170( 
2. 80300 ( 
3. 39220 ( 
2.66130( 
4.33860 ( 
5. 79990 ( 
3 . 63520 ( 
4.90370 ( 
5.91410( 
5. 46220 ( 


.135125- 
.071750- 
.019483- 
.038037- 
.018531- 
.074085- 
.023017- 
.011203- 
. 050924  - 
.024675- 
.021932- 
.045010- 
.024739- 
.038272- 
.054735- 
.0.39516- 
.044718- 
.027492- 


a  ) 

■a') 
■a') 
■a') 
■a') 
a') 
-a') 
■a') 
■a') 
■a') 
■a') 
■a') 
■a') 
■a') 
-a') 
-a') 
-a') 
■a') 


Note. — Values  of  R  and  log  C  are  taken  from  the  results  of  work  in  Example  7, 
Chapter  8. 


PEACTICAL  METHODS 


119 


4.  For  the  trajectories  given  in  the  preceding  example  (Example  3)  compute 
the  abscissa  and  ordinate  of  the  vertex ;  and  the  ordinate  and  inclination  of  the  curve 
at  each  of  the  points  whose  abscissa  are  given  below,  and  also  at  the  point  of  fall. 


ANSWERS. 

£ 

Vertex. 

Point  of  fall. 

For  different  abscissae. 

Xo- 

yo=Y. 

I'o- 

e=- 

Xi. 

2/1- 

Oi. 

X2. 

2/2. 

P^ 

Yds. 

Feet. 

Feet. 

—  CtJ. 

Yds. 

Feet. 

Yds. 

Feet. 

02 

A... 

1112 

175 

0. 

(— )  6° 

45.0' 

750 

154.5 

2° 

02.9' 

1800 

96.7 

(— )  4° 

24.8' 

B... 

2160 

200 

0.08 

(-)  5 

32.4 

1000 

133.0 

2 

00.5 

3000 

149.2 

(— )  2 

29.4 

c... 

2056 

100 

0. 

(-)  2 

19.9 

1000 

71.7 

0 

59.2 

3000 

73.7 

(— )  1 

07.2 

D... 

2476 

163 

0.05 

(— )  3 

44.4 

1000 

98.3 

1 

31.9 

3000 

152.7 

(— )  0 

46.5 

E... 

2414 

120 

0.64 

(-)  2 

24.7 

1000 

75.8 

1 

07.8 

3000 

111.2 

(— )  0 

34.8 

F... 

7.329 

2534 

(-)1.61 

(-)19 

38.2 

4000 

1910.7 

0 

34.8 

9000 

2332.6 

(— )  4 

50.6 

G... 

2097 

115 

0.49 

(-)  2 

40.6 

1000 

81.1 

1 

08.1 

3000 

87.9 

(— )  1 

11.9 

H.. 

1SS8 

80 

0.03 

(-)  1 

50.8 

1000 

61.1 

0 

47.3 

3000 

48.1 

(— )  1 

07.7 

I... 

4102 

020 

0.37 

(-)  8 

30.6 

2000 

433.7 

3 

11.3 

4000 

618.9 

(+)  0 

11.2 

J... 

3990 

450 

0.19 

(-)  5 

29.1 

2000 

328.0 

2 

14.3 

4000 

448.6 

0 

00.0 

K.. 

4511 

544 

0.43 

(-)  5 

46.7 

2000 

388.1 

2 

37.3 

4000 

535 . 9 

(+)  0 

35.7 

L... 

5709 

1184 

0.17 

(-)ll 

08.4 

3000 

869.7 

4 

00.3 

9000 

029.4 

(— )  7 

08.8 

M.. 

6140 

1075 

(— )0..30 

(-)  8 

29.0 

3000 

769.4 

3 

33.0 

9000 

770. -2 

(— )  4 

12.5 

N.. 

121.54 

4.581 

0.38 

(-)19 

55.7 

9000 

4209.5 

4 

20.3 

19000 

2360.4 

(— )13 

14.0 

0... 

5675 

1873 

(-)1.47 

(-)lfi 

10.0 

3000 

1411.2 

0 

15.0 

9000 

980.3 

(— )io 

47.5 

P... 

5970 

1S79 

0.33 

(-)14 

44.2 

3000 

1376.5 

6 

11.5 

9000 

1272.3 

(— )  8 

02.4 

Q... 

7714 

3339 

(-)0.25 

(— )20 

15.0 

3000 

2000.0 

10 

06.3 

9000 

3220.6 

(— )  3 

28.2 

R... 

7810 

1915 

0.47 

(-)ll 

54.7 

3000 

1141.0 

5 

47.7 

9000 

1857.2 

(-)  1 

48.8 

Note. — Of  course,  the  found  values  of  the  ordinate  and  of  the  angle  of  Incllnatloii 
at  the  point  of  fall  should  be  numerically  equal  to  zero  and  to  the  angle  of  fall,  respectively. 
The  actual  results  obtained  by  the  above  work  show  the  degree  of  accuracy  of  the  method. 


CHAPTER  10. 

THE  DERIVATION  AND  USE  OF  SPECIAL  FORMULA  FOR  FINDING  THE 
HORIZONTAL  RANGE,  TIME  OF  FLIGHT,  ANGLE  OF  FALL,  AND  STRIK- 
ING VELOCITY  FOR  A  GIVEN  ANGLE  OF  DEPARTURE  AND  INITIAL 
VELOCITY. 

177.  For  this  problem  we  use  expressions  that  have  already  been  derived, 
namely,  equations  (113),  (115),  (116),  (117)  and  (118);  in  some  cases  somevs^hat 
transposed,  as  follows  (neglecting  the  altitude  factor)  : 

A^'-^    '  (156) 

X  =  CZ  (157) 

tana,  =  i?'tan<^  (158) 

T  =  CT' sec  cf>  (159) 

t?tj  =  Ujj  cos  ^  sec  (0  (160) 

178.  As  no  account  is  taken  of  the  altitude  factor  in  the  above  expression,  we 
cannot  use  our  standard  problem  at  present,  so  we  will  take  a  different  case  for  our 
first  solution,  and  one  at  such  a  short  range  that  the  altitude  factor  may  be  neglected 
without  material  error.  Let  us  therefore  compute  the  values  of  X,  w,  T  and  v^  for 
an  angle  of  departure  of  1°  02'  34",  for  the  5"  gun  for  which  y  =  3150  f.  s.,  w  =  50 
pounds  and  c  =  0.61,  for  a  barometer  reading  of  30.00"  and  a  thermometer  reading 
of  50°  F. 

From  Table  VI. 

E=    log  0.51570 

8=  1.035    loff  0.01494 


C=    log  0.50076 colog  9.49924-10 

2(^  =  2°  04'  48"    sin  8.55984-10 

4  =  .01146    log  8.05908-10 

From  which,  from  Table  II,  we  get 

Z  =  2700+   nnnLf'^'^^n^n^^"^  +  .01146  -  .01119')  =  2700  + 121.1  =  2821.1 
log  B'  =  .0945  +  :00||8M  -  .521|p  =  .09449 
«„  =  3235-  ^^-j^  +  52^  =3270.8 
r  =  1.064+  ^^  -  :M5X50  ^j^„.^Q 

J.  V  \J  J.UU 


PEACTICAL  METHODS.  121 

C=    log  0.50076 log  0.50076 

Z  =  2821.1    log.  3.45042 

<^  =  1°02'24"    tan  8.25894-10.  .sec  0.00007.  .cos  9.99993-10 

i?'=    log  0.09449 

^^.  =  2270.8    log  3.35618 

r  =  1.056    loff  0.02366 


X  =  8936.7    los  3.95118 


.  =  1°   17'  34"    tan  8.35343-10 sec  0.00011 


r  =  3.3457    loff  0.52449 


ra,  =  2271    log  3.35622 

Z  =  2978.9  yards.  T  =  3.3457  seconds. 

(0=1°   17'  34".  t'<^  =  2271  f.  s. 

179.  Or,  with  perhaps  no  more  labor,  we  may  avoid  the  double  interpolation 
necessary  in  the  above  solution  by  working  the  problem  for  y  =  3100  f.  s.  and  then 
again  for  7=:  3200  f.  s.,  and  then  get  our  final  results  by  interpolation  between 
those  obtained  for  the  two  velocities.  In  this  case,  as  7  =  3150,  our  final  results 
should  be  half  way  between  the  results  obtained  for  the  two  values  of  V  with  which 
we  work.  The  value  of  C  and  that  of  A  are  of  course  the  same  as  in  the  preceding 
problem,  so  starting  from  that  point,  with  J.  =  .01146  and  log  C  =  0.50076,  we  have 

For  F  =  3100  f.  s. 

Z  =  2751.9         logS'  =  .09282         ^^.  =  2248.5         r  =  1.0428 

C=    log  0.50076 log  0.50076 

Z  =  2751.9    log  3.43963 

(>  =  1°02'24"    tan  8.25894-10.  .sec  0.00007.  .cos  9.99993-10 

B'=     log  0.09282 

«^  =  2248.5    log  3.35190 

T'  =  1.0428    log  0.01820 


X  =  8717.4    log  3.94039 


.  =  1°  17' 16"    tan  8.35176-10 sec  0.00011 


r  =  3.3039    log  0.51903 


r^  =  2248.75    log  3.35194 

For  7  =  3200  f.  s. 
Z  =  2904         log5'  =  .09674         m„  =  2288.9         T'=1.0738 

C=    log  0.50076 log  0.5007G 

Z  =  2904    log  3.46300 

c>  =  l°  02'  24"    tan  8.25894-10.  .sec  0.00007.  .cos  9.99993-10 

B'=     log  0.09674 

w<,  =  2288.9    log  3.35963 

r  =  1.0738    log  0.03092 


X  =  9199.4    lo?  3.96376 


w  =  l°17'5S"    tan  8.35568-10 sec  0.00011 

r  =  3.4021    log  0.53175 

r«  =  2289.1    log  3.35967 


122  EXTEEIOE  BALLISTICS 

Our  results  then  are 

For  7  =  3100  f .  s.  For  F  =  3200  f .  s.  For  7  =  3150  f .  s. 

(By  interpolation  be- 
tween the  results 
obtained  for  3100 
and  3200  f.  s.) 

Z   2905.8  yards.  3066.5  yards.  2986.1  yards. 

^    1°   17'  16".  1°   17'  57".  1°   17'  37". 

T   3.3039  seconds.  3.4022  seconds.  3.3531  seconds. 

v^    2248.75  f .  s.  2289.1  f .  s.  2268.9  f .  s. 

180.  We  will  now  take  our  standard  problem,  introduce  the  altitude  factor,  and 
solve.  This  for  the  12"  gun,  for  which  7  =  2900  f.  s.,  w  =  870  pounds  and  c  =  0.61. 
For  this  problem  we  will  take  the  angle  of  departure  as  4°  13'  14",  which  we  already 
know  corresponds  to  a  range  of  10,000  yards,  and  will  consider  the  atmospheric  con- 
ditions as  standard.  Proceeding  in  a  manner  similar  to  that  employed  in  originally 
computing  the  angle  of  departure,  that  is,  by  performing  the  work  first  without  con- 
sidering /  until  we  have  gone  far  enough  to  enable  us  to  determine  the  value  of  /  by 
a  series  of  approximations,  and  then  introducing  it,  we  have,  by  the  use  of  the  formulae 
employed  in  the  preceding  problem,  and  in  addition  of 

F  =  1"C  tan  </,     and     loglog/  =  log  F  +  5.01765-10 

Ci  =  Z=  (from  Table  VI)    colog  9.00417-10 

2<;6  =  8°  26'  28"    sin  9.16670-10 

«,;  =  .014821   log  8.17087-10 

^^.^3,^^.000021X57  ^3,Q 

A/'  =  850    log  2.92942 

(J  -    log  0.99583 

<^  =  4°   13'  14"    tan  8.86803-10 

y  -  log  2.79328 

Constant    log  5.01765-10 

f^^    log  0.00647 loglog  7.81093-10 

c\=    log  0.99583 

C^=    log  1.00230 colog  8.99770-10 

2<^  =  8°  26'  28" sin  9.16670-10 

ao;  =  .014602   log  8.16440-10 

4,"  =  793+:^^-^-|<^^  =838.92 

A;'  =  838.92   log  S.92372 

(j^^    log  1.00230 

^'_40   -i^g.  -^^n   tan  8.86803-10 

Y  =    log  2.79405 

Constant    log  5.01765-10 

f^.^    log  0.00648 loglog  7.81170-10 

C\=    log  0.99583 

(73=    log  1.00231 colog  8.99769-10 

2<^  =  8°  26'  28"    sin  9.16670-10 

a,;  =  .014601    log  8.16439-10 


PEACTICAL  METHODS  123 

A,"  =  7d3-\-  '^^^Qoli^^^^ 838.87 

^3"  =  838.87   log  2.92370 

Cs=    log  1.00231 

<j>  =  4:°   13'  14"    tan  8.86803-10 

1^3=     log  2.79404 

Constant    loir  5.01765-10 


/3=     log  0.00648 logiog  7.81169-10 

C,=    W  0.99583 


C,= log  1.00231 

and  as  0^  =  0^,  we  see  that  we  have  reached  the  limit  of  accuracy  in  determining  the 
value  of  C,  and  we  therefore  proceed  with  the  work  with  log  (7=1.00231,  4=ao' 
=  .014601,  F  =  2900,  and  from  Table  II  Z  =  2984,  log  i3'  =  . 10371,  r  =  1.2332  and 
t/^^  =  2026.2.    The  further  work  then  becomes 

C=    log  1.00231 log  1.00231 

Z  =  2984    log  3.47480 

<;!>  =  4°   13'  14"    tan  8.86803-10.  .sec  0.00118.  .cos  9.99882-10 

B'=     log  0.10371 

r  =  1.2332    log  0.09103 

Wa,  =  2026.2    W  3.30668 


2:  =  29999    loir  4.47711 


>  =  5°  21'  11"    tan  8.97174-10 sec  0.00190 


T  =  12.431    locr  1.09452 


t'^  =  2029.55    log  3.30740 

A  comparison  between  these  results  and  those  obtained  in  Chapter  8,  where  we 
computed  the  values  of  the  same  elements  with  V  and  X  as  the  data,  gives  an  inter- 
esting measure  of  the  accuracy  of  the  methods  employed.  Tabulating  these  results 
for  comparison,  we  have: 

Value  by  work  under 

Element.  Chapter  8.  This  Chapter. 

R 10,000  yards.  10,000  yards. 

CO 5°  21'  11".  5°   21'  11". 

T 12.43  seconds.  12.431  seconds. 

Vu 2029.0  foot-seconds.  2029.6  foot-seconds. 


124 


EXTERIOR  BALLISTICS 
EXAMPLES. 


1.  Given  the  data  contained  in  the  following  table,  compute  the  values  of  R,  w, 
T  and  v^,  by  Ingalls'  methods,  using  Table  II,  and  determining  the  value  of  /  by- 
successive  approximations,  and  applying  it  to  get  the  correct  value  of  the  ballistic 
coefficient. 


DATA. 

ANSWERS. 

Prob 
lem. 

Projectile. 

Atmosphere. 

Ve- 
locity. 

f.  s: 

0. 

Range. 

Yds. 

u. 

T. 
Sees. 

Vo3. 
f.  8. 

a. 

If. 

Bar. 

Ther. 

In. 

Lbs. 

In. 

°F. 

A.... 

3 

13 

1.00 

28.00 

0 

1150 

7° 

13' 

36" 

2564 

8° 

59' 

8.22 

816 

B.... 

.        3 

13 

1.00 

28.10 

^ 

2700 

3 

45 

36 

4072 

7 

18 

8  22 

988 

C... 

4 

33 

0.67 

28.50 

10 

2900 

1 

35 

36 

3547 

2 

09 

4.61 

1850 

D.... 

5 

50 

1.00 

28.67 

15 

3150 

2 

02 

42 

4088 

3 

29 

5.90 

1412 

E.... 

.5 

50 

0.61 

29.00 

20 

3150 

1 

30 

12 

3934 

2 

02 

4.71 

2004 

F.... 

6 

105 

0.61 

29.83 

25 

2600 

12 

30 

06 

13201 

22 

36 

26.98 

1067 

G.... 

6 

105 

1.00 

29.75 

35 

2800 

1 

48 

36 

3646 

2 

30 

5.01 

1716 

H... 

.        6 

105 

0.61 

30.00 

43 

2800 

1 

00 

00 

24H6 

1 

08 

2.94 

2306 

I.... 

.        7 

105 

1.00 

30.20 

47 

2700 

3 

59 

54 

6189 

6 

28 

10.06 

1311 

J.... 

.       7 

165 

0.61 

30.50 

51 

2700 

3 

00 

24 

5929 

3 

58 

8.15 

1781 

K... 

8 

260 

0.61 

.30.75 

58 

2750 

3 

31 

54 

7214 

4 

39 

9.73 

1773 

L.... 

.      10 

510 

1.00 

31.00 

65 

2700 

6 

17 

48 

9665 

10 

10 

15.81 

1313 

M... 

.      10 

510 

0.61 

30.45 

75 

2700 

5 

43 

54 

10806 

7 

50 

15.34 

1692 

N  ... 

.      12 

870 

0.61 

30.20 

80 

2900 

10 

45 

00 

19751 

16 

32 

29.40 

1516 

0.... 

.      13 

1130 

1.00 

30.00 

85 

2000 

10 

12 

24 

9810 

14 

20 

19.73 

1205 

P.... 

.      13 

1130 

0.74 

29.50 

90 

2000 

10 

40 

48 

11086 

14 

06 

21.16 

1315 

Q.... 

.      14 

1400 

0.70 

29.00 

95 

2000 

13 

45 

18 

1.3838 

18 

17 

27.01 

1302 

E.... 

.      14 

1400 

0.70 

28.67 

100 

2600 

8 

19 

00 

14553 

11 

14 

21.44 

1657 

2.  Given  the  data  contained  in  the  following  table,  compute  the  values  of  R,  w, 
T  and  f ^  by  Ingalls'  methods,  using  Table  II,  and  correcting  for  /  by  the  use  of  the 
maximum  ordinate  given  in  the  table. 


A.. 
B.. 
C. 
D.. 
E.. 
F.. 
G.. 
H. 
I.. 
J.  . 
K. 
L.. 
M. 
N  . 
O  . 
P.. 
Q.. 
R.. 


DATA. 


Projectile. 

Atmosphere. 

Ve- 
locity. 

d. 

10. 

c. 

Bar. 

Ther. 

f.s. 

In. 

Lbs. 

In. 

°F. 

3 

13 

i.do 

28.10 

100 

1150 

3 

13 

1.00 

28.40 

95 

2700 

4 

33 

0.67 

28.25 

90 

2900 

5 

50 

1.00 

29.00 

87 

3150 

5 

50 

0.61 

29.20 

85 

3150 

6 

105 

0.61 

29.50 

80 

2600 

6 

105 

1.00 

29.75 

77 

2800 

6 

105 

0.61 

30.00 

73 

2800 

< 

165 

1.00 

30.10 

70 

2700 

7 

165 

0.61 

30.20 

60 

2700 

8 

260 

0.61 

31.00 

50 

2750 

10 

510 

1.00 

30.50 

40 

2700 

10 

510 

0.61 

30.30 

35 

2700 

12 

870 

0.61 

30.00 

31 

2900 

13 

1130 

1.00 

29.75 

20 

2000 

13 

1130 

0.74 

29., 50 

22 

2000 

14 

1400 

0.70 

29.00 

20 

2000 

14 

1400 

0.70 

28.50 

10 

2600 

14' 
09 


02 
55 


41 
05 
03 
11 
55 
18 
44 
54 
45 
05 
40 


18" 
24 


06  06 
27   42 


24 
18 


18   48 


00 
48 
48 
18 
36 
24 
18 
12 
12 
00 
18 


Maxi- 
mum 
ordi- 
nate. 
Feet. 


150 

115 

43 

79 

45 

2481 

L59 

95 

638 

471 

532 

1214 

824 

5363 

1488 

1511 

3450 

1569 


ANSWERS. 


Ranee 
Yds. 


2042 
3245 
2757 
3479 
3035 

12755 
4447 
3912 
7298 
7449 
81.32 

10073 
9991 

21989 
9220 
9978 

14043 

13017 


T. 

Sees. 

6° 

08' 

6.14 

3 

35 

5.38 

1 

20 

3.29 

2 

09 

4.46 

1 

16 

3.36 

19 

16 

24.. 58 

3 

20 

6.32 

2 

03 

4.86 

8 

42 

12.50 

5 

43 

10.79 

5 

46 

11.38 

11 

37 

17.08 

7 

16 

14.19 

22 

52 

35.96 

14 

13 

18.94 

13 

05 

19.24 

21 

01 

28.92 

10 

40 

19.60 

Vol, 

f.  s. 


900 
1263 
2179 
1759 
2333 
1115 
1616 
2094 
1213 
1619 
1702 
1245 
1685 
1345 
1167 
1284 
1219 
1585 


CHAPTEE  11. 

THE  DERIVATION  AND  USE  OF  SPECIAL  FORMULiE  FOR  FINDING  THE 
ANGLE  OF  ELEVATION  NECESSARY  TO  HIT  A  POINT  ABOVE  OR  BELOW 
THE  LEVEL  OF  THE  GUN  AND  AT  A  GIVEN  HORIZONTAL  DISTANCE 
FROM  THE  GUN,  AND  THE  TIME  OF  FLIGHT  TO  AND  REMAINING 
VELOCITY  AND  STRIKING  ANGLE  AT  THE  TARGET;  GIVEN  THE 
INITIAL  VELOCITY. 

New  Symbols  Introduced. 

(f>x. . '  .Angle  of  departure  for  a  horizontal  distance  x. 
181.  As  we  know,  the  secondary  function  a  refers  to  the  point  (x,  y)  in  the   Transfor- 

'  "^  \  •  T        mation  of 

trajectory  (A  being  its  special  value  when  x  =  X  and  y  =  0) ;  but  since  the  pseudo   formulae. 

velocity  is  independent  of  the  height  of  this  point,  and  dependent  only  on  x,  or  the 

horizontal  distance  from  the  muzzle  of  the  gun,  we  may  consider  x  as  the  horizontal 

range  of  another  trajectory  having  the  same  initial  velocity,  whose  angle  of  departure 

may  be  designated  by  cjix.    For  this  case  then,  we  have,  from  (115) 

_  sin  2(^j 
a-       ^~ 


(161) 


as  well  as  A=^^^  (163) 

182.  We  have  also  derived,  in  (141),  the  expression 

y-=  --^  (A-a)  (163) 

X       2  cos^  <^  ^  ^  ^       ^ 

183.  Now  substituting  in  (163)  from  (161)  and  (162),  we  get 

y  _        C      /sin  2(f>  _  sin  2cf).A 
~x~  2  cos""  <I>[~~C~~  C~l 

y  _  sin  2<^  — sin  24>x 
X  2  cos^  <j> 

184.  iSTow  if  p  be  the  angle  of  position,  we  know  that 

tan  p=  ^--  (IGo) 


(164) 


X 


and  combining  (164)  and  (165)  we  get 

sin2^-sin^. 
'  2  cos^  </) 

or  sin  2<^x  =  sin  2^  — 2  cos- 0  tan /?  (1G6) 

but  as  2  cos-  <;6  =  l  +  cos  2<^  (166)  becomes 

sin  2<^j.  =  sin  2<^  — tan  /)(l4-cos  2<f)) 
or  sin  2<f>  —  cos  2<f)  tan  ;;  —  tan  ;;  =  sin  2<^j, 

Multiplying  and  dividing  the  first  member  by  cos  p  gives 

sin  2<^  cos  ;j  — cos  2<A  sinp  — sin  w       .    ^, 

* —  —  sm  4<px 

cos  p 

sin(2d>  — /;)  —  sin  »        •    ^^ 

or  - — >^— c — i^ ^  =sm2<bx 

cosp 

whence  sin  ( 2<^  —  p )  =  sin  /)  +  cos  p  sin  2<^x 

or  sin(2<^-/^)=sin/;(l  +  cot/)sin2<^x)  (167) 

And  we  also  know  that  \l/  =  cj>  —  p. 


Elevated 
target. 


126  EXTERIOR  BALLISTICS 

185.  Now  for  any  given  point  (x,  y)  we  may  compute  the  value  of  z  from 
z=  ^  ,  and  then,  with  z  and  7. as  arguments,  we  may  take  the  value  of  a  from  the 

A  column  of  Table  II. 

186.  We  may  then  compute  the  value  of  sin  2</>x  from  sin  2(f>x=  —aC  and  thence 
of  ^  from  (167).  Also,  with  the  same  arguments,  we  may  take  from  Table  II  the 
values  of  a',  u  and  t'  for  the  point  (x,y). 

187.  Assembling  the  formulae  deduced  in  this  chapter,  and  also  the  other 
necessary  formulae  previously  deduced  as  given  in  (85),  (113),  (138)  and  (1-14), 
we  have,  for  the  solution  of  this  problem, 

tanp=y-  (169) 


(170) 


sm2cf>a;  =  aC  (171) 

sin(3<^-;5)  =sin  p(l  +  cot  p  sin  2cf>a;)  (172) 

^  =  ct>-p  (173) 

A='-^  (174) 

tau^=;^^  (.1-a')  ■  (175) 

^  =  Cfsec</>  (176) 

t; z=  M  cos  ^  sec  ^  (!''"<') 

.  188.  Let  us  now  compute  these  several  elements  for  our  standard  problem 
12"  gun,  y  =  2900  f.  s.,  w=:870  pounds,  c  =  0.61,  for  a  target  at  a  horizontal  distance 
of  10,000  yards  from  the  gun,  and  1500  feet  above  the  level  of  the  gun,  the  barometer 
being  at  29.00"  and  the  thermometer  at  90°  F.  Also,  instead  of  computing  the  alti- 
tude factor,  we  will  take  it  from  Table  V  as  being  sufficiently  accurate  for  this 
purpose. 

Taking  the  mean  altitude  as  two-thirds  of  1500  feet,  that  is,  1000  feet.  Table  V 
gives  us  that  /=  1.026.    Taking  K  from  Table  VI. 

K=    log  0.99583 

/=1.026    log  0.01115 

S  =  .921    log  9.96426-10 colog  0.03574 

C=    log  1.04272 colog  8.95726-10 

?/  =  1500    log  3.17609 

a-  =  30000    log  4.47712 log  4.47712 


p  =  2°   51'  45"    log  8.69897-10 


!  =  2719.0    log  3.43440 


PRACTICAL  METHODS  127 

From  Table  II 

a=.01299         a'  =  Md2         w  =  2095.1         f  =  1.104 

C=    log  1.04272 

a  =  .01299    log  8.11361-10 

2<^,,=    sin  9.15633-10 

p  =  2°   51'  45"    cot  1.30-103 

cot  p  sin  2(^^  =  2.8665    log  0.45736 

1  +  cot ;?  sin  2cf>^  =  3.8665    log  0.58732 

j)  =  2°  51'  45"    sin  8.69844-10 

2(p-p  =  ir  07'  59"    sin  9.28576-10 

p=   2°   51'  45"  cf>  =  6°  59'  52" 

20=13°   59'  44"  p  =  2°   51'  45" 

cf>=   6°   59'  52"  ip  =  4:°  08'  07" 

2(^  =  13°   59'  44"    sin  9.38356-10 

C=    colog  8.95728-10 


A=         .02192    log  8.34084-10 

a'=         .02920 


A-a'=(-).0072S    (-)log  7.86213-10 

(f,  =  6°  59'  52"    tan  9.08908-10.  .sec  0.00324. . .     cos  9.99676-10 

A  =  .02192    eolog  1.65916 

r=1.104    log  0.04297 

u  =  2095A    log  3.32120 

C=    log  1.04272 

^=(-)2°  20'05"    ..(-)tan  8.61037-10 sec  0.00036 

^  =  12.273    log  1.08893 


v  =  2081.2    log  3.31832 

189.  Xow  suppose  that,  instead  of  the  conditions  worked  out  above,  the  gun  had   Depressei 
been  in  a  battery  on  the  hill  and  the  ship  had  been  the  target,  all  other  conditions     "^®  ' 
being  the  same.    The  work  would  have  been  the  same  down  to  and  including  the 
determination  of  the  values  of  a,  a',  u  and  f,  except  that  y  is  negative,  and  therefore 
p={  —  )2°  51'  45".    We  then  proceed  as  before,  but  with  this  negative  value  of  p 
instead  of  the  positive  one  employed  before,  and  the  subsequent  work  becomes : 

cot  p  sin  2<j>x  =  ( —  )  2.8665 

l  +  cot/Jsin2</>^=(-)1.8665    (-)log  0.27103 

p={-)2°  51'  45"    (-)sin  8.69844-10 

2cl>-p  =  o°   19'  32"    (  +  )sin  8.96947-10 

p=(-)2°   51'  45"  <l>=  1°   13'  54" 

2<f>=         2°   27'  47"  p=(-)2°   51'  45" 

«;!,=  1°   13'  54"  ij;=  4°   05'  39" 


128 


EXTERIOR  BALLISTICS 


2<^  =  2°  27'  47"    sin  8.63321-10 

C=    coloff  8.95728-10 

A  = 

a!  — 


.00390    log  7.59049-10 

.02920 


A-a'=(-).02530    (-)log  8.40312-10 

<^  =  1°13'54"    tan  8.33243-10.  .sec  0.00010.  . .     cos  9.99990-10 

A  =  .0039    colog  2.40951 

f  =  1.104    log  0.04297 

w^2095.1    log  3.32120 

C=    loff  1.04272 


'=(-)7°  57'01"    ..(-)tan  9.14506-10, 


sec  0.00419 


^  =  12.184    W  1.08579 


i'  =  2114.9    log  3.32529 

190.  Assembling  the  results  of  these  last  two  problems  for  comparison,  we  have : 

Value  for 
Ship  attacking  battery.  Battery  attacking  ship. 

t^' 4°  08'  07".  4°  05'  39". 

6 i-)2°  20'  05".  (-)7°   57'  01". 


t 12.273  seconds. 

V 2081.2  f.  s. 


12.184  seconds. 
2114.9  f.  s. 


In  working  problems  similar  to  the  above,  great  care  must  be  taken  to  carry 
through  consistently  the  signs  of  the  several  quantities  and  logarithms. 


Figure  14. 


191.  From  Figure  14,  in  which  (a)  represents  the  first  case  and  (&)  the  second, 
we  plainly  see  that  in  (a)  the  force  of  gravity  acts  to  reduce  the  velocity  of  the  pro- 
jectile, and  in  (h)  to  increase  it  from  what  it  would  be  in  the  horizontal  trajectory. 
Therefore  we  would  expect  to  find  it  necessary  to  give  the  gun  a  greater  elevation 
relative  to  the  line  of  sight  in  order  to  hit  in  (a)  than  in  (b),  and  the  results  of  the 
work  show  that  such  is  the  case. 

192.  Also,  from  the  figures  we  can  see  that  the  angle  of  inclination  of  the  curve 
to  the  horizontal  at  the  point  of  impact  would  be  greater  in  (&)  than  in  (a),  which 
is  again  shown  by  the  work. 


PEACTICAL  METHODS  129 

193.  Also,  as  gravity  in  (o)  reduces  and  in  (&)  increases  the  velocity,  we  would 
expect  to  have  the  remaining  velocity  less  and  the  time  of  flight  greater  in  (a)  than 
in  (b),  and  again  the  work  shows  this  to  be  the  case. 

194.  The  angle  of  elevation  resulting  from  the  work  is  of  course  the  angle  at 
which  the  gun  must  be  pointed  above  the  target  in  either  case,  that  is,  above  the  line 
of  sight  AB.  The  sight  drums  are  marked  in  yards,  however,  and  not  in  degrees  of 
elevation ;  so  to  practically  set  the  sights  we  look  in  the  range  table  of  the  gun  and  find 
in  Column  2  an  angle  of  departure  equal  to  our  found  angle  of  elevation,  and  find  in 
Column  1  the  range  in  yards  corresponding  to  that  angle  of  departure.  We  then  set 
our  sights  in  range  to  that  number  of  yards,  and  point  the  gun  at  the  target,  that  is, 
bring  the  line  of  sight  to  coincide  with  the  line  AB  of  the  figure.  The  gun  is  then 
elevated  at  the  proper  angle  above  the  line  AB,  \p  from  the  work,  and  at  an  angle  of 
departure  above  the  horizontal  of  <l>  =  xp-\-p. 

195.  In  the  problem  shown  in  Figure  14(a),  we  have  by  a  simple  interpolation 
between  Columns  1  and  3,  that  the  range  corresponding  to  an  angle  of  departure  of 
4°  08'  07"  is  9841  yards,  which  is  the  range  at  which  the  sight  should  be  set. 

196.  Similarly,  in  Figure  14(6),  the  sight  should  be  set  for  an  angle  of 
departure  of  4°  05'  39",  that  is,  at  9764  yards. 

197.  In  Chapters  8,  9  and  10,  and  in  this  chapter,  we  have  shown  the  methods 
and  formula  to  be  employed  in  solving  certain  of  the  more  common  and  more 
important  ballistic  problems.  Those  selected  for  the  purposes  of  this  book  are  the 
ones  most  likely  to  be  encountered  in  naval  practice,  but  there  are  a  large  number 
of  others  that  may  arise  under  special  circumstances,  which  may  be  solved  by  similar 
methods.  Some  of  the  more  important  of  these  are  enumerated  below,  to  show  the 
scope  of  the  methods  that  have  been  taught,  for  they  are  all  solved  in  similar  ways. 
In  each  case  the  solution  consists  of  a  preliminary  transformation  of  the  fundamental 
ballistic  formula,  in  a  manner  similar  to  those  shown  in  the  preceding  pages  of  this 
book,  in  order  to  fit  them  for  use  in  the  particular  problem  under  consideration ;  and 
then  the  necessary  computations  may  be  made  from  the  resultant  equations.  It 
should  also  be  borne  in  mind  that  all  our  work  has  so  far  applied  only  to  direct  fire,  as 
do  also  the  problems  enumerated  below,  and  that  when  problems  incident  to  mortar 
fire  and  other  special  classes  of  work  are  added,  the  number  of  problems  that  may 
present  themselves  becomes  very  large.  Beside  the  problems  already  explained  in 
these  pages,  some  of  the  simpler  direct  fire  problems  that  may  be  readily  solved  by 
similar  methods  are: 

(a)  Knowing  X,  C  and  Va,;  to  compute  V. 

(b)  Knowing  V,  C  and  Voj',  to  compute  X. 

(c)  Knowing  V,  X  and  C;  to  compute  T. 

(d)  Knowing  V,  T  and  C;  to  compute  Vu. 

(e)  Knowing  V,  X  and  0;  to  compute  C. 

(f)  Knowing  V,  C  and  v^;  to  compute  X,  <j>,  w  and  T, 

(g)  Knowing  V,  C  and  w ;  to  compute  X,  <f>,  T  and  Vo). 
(h)  Knowing  X,  <f>  and  C;  to  compute  V. 

(i)   Knowing  T,  </>  and  C;  to  compute  V. 


130 


EXTERIOE  BALLISTICS 


EXAMPLE. 

1.  Given  the  data  contained  in  the  following  table,  compute  the  values  of  ip,  t, 
V  and  6  for  the  given  values  of  x  and  ij,  both  when  ij  is  positive  and  when  it  is 
negative ;  and,  whenever  the  range  tables  available  permit,  tell  how  to  set  the  sight  in 
elevation  in  each  case  in  order  to  hit.    Correct  for  /  from  the  data  given. 


Problem. 


DATA. 


Projectile. 


d. 
In. 


A 

B 

C 

D 

E 

F 

G 

H 

I. 

J 

K 

L 

M 

N 

0 

P 

Q 

R 


3 

3 

4 

5 

5 

6 

6 

6 

7 

7 

8 

10 

10 

12 

13 

13 

14 

14 


w. 
Lbs. 


13 

13 

33 

50 

50 

105 

105 

105 

165 

165 

260 

510 

510 

870 

1130 

1130 

1400 

1400 


1.00 
1.00 
0.67 
1.00 
0.61 
0.61 
1.00 
0.61 
1.00 
0.61 
0.61 
1.00 
0.61 
0.,61 
1.00 
0.74 
0.70 
0.70 


Atmosphere. 


Bar. 
In. 


31.00 
30.90 
30.75 
30.33 
30.00 
29.80 
29.50 
29.25 
29.00 
28.50 
28.25 
28.00 
28.10 
28.07 
29.00 
29.15 
29.75 
30.00 


Ther. 
°F. 


5 
10 
20 
27 
33 
37 
32 
30 
40 
50 
55 
60 
75 
80 
83 
87 
93 
97 


Ve- 
locity 
f.  s. 


1150 
2700 
2900 
3150 
31.50 
2600 
2800 
2800 
2700 
2700 
2750 
2700 
2700 
2900 
2000 
2000 
2000 
2600 


Chart  dis- 
tance from 
gun  to 
target. 
Yds. 


2500 

4000 

3300 

4200 

3700 

7300 

3800 

3100 

6800 

7200 

7800 

9700 

10800 

20500 

10500 

11000 

13000 

13500 


Height 

of  target 

above  or 

below  gun. 

Feet. 


-  200 
:  350 
:  400 
:  450 

-  475 

-  600 
:  500 

-  460 

-  7.50 
:  800 

-  825 

-  850 
:  900 
-1500 
-1100 
■1000 

-  950 
-1200 


Maximum 
ordinate  for 

trajectory 

of  range  x. 

Feet. 


252 

251 

70 

144 

75 

522 

109 

57 

518 

424 

456 

995 

952 

3994 

1937 

1830 

2637 

1630 


ANSWERS. 

Prob- 
lem. 

When  y  is  positive. 

When  y  is  negative. 

^P. 

Set 
sight 

at: 
Yds. 

e. 

t. 

Sees. 

V. 

f.  s. 

i. 

Set 
sight 

at: 
Yds. 

e. 

t. 

Sees. 

V. 

f.  s. 

A 

B 

C 

D 

E 

F 

G 

H  

I 

J 

K 

L 

M 

N 

0 

P 

Q 

R 

7°  11.4' 

3  56.9 

1  28.3 

2  11.0 
1  23.3 

4  38.8 
1  54.9 
1  17.5 

4  36.0 

3  49.1 
3  49.0 

6  02.2 

5  35.5 
11  14.3 

11  15.6 
10  39.1 

12  54.0 

7  42.8 

4308 

3388 

4364 

3764 

7452 

3854 

3117 

6869 

7170 

7744 

9543 

10G04 

20180 

10373 

10879 

13353 

{—)   7°  34.4' 

(_)  6  09.4 

0  20.3 

(— )  1  48.3 

0  36.4 
(— )  5  25.1 
(— )  0  11.6 

1  18.5 
(— )  5  37.5 
(— )  3  07.8 
(— )  3  01.5 
(— )  7  46.7 
( — )  5  55.3 
(— )16  02.3 
(— )14  13.5 
(— )12  30.4 
(— )15.57.0 
(— )  8  48.8 

8.13 

8.38 

4.27 

6.24 

4.38 

11.60 

5.29 

3.77 

11.42 

10.23 

10.54 

15.40 

15.09 

30.72 

21 .  62 

21.12 

25.39 

19.91 

795 
948 
1866 
1345 
2059 
1412 
1679 
2192 
1244 
1675 
1815 
1377 
1741 
1502 
1169 
1299 
1287 
1645 

7°  08.8' 

3  56.0 

1  28.1 

2  10.6 
1  22.9 

4  37.5 
1  54.6 
1  17.3 

4  34.4 

3  48.0 
3  47.9 

5  59.9 
5  33.6 

11  07.5 
11  05.8 
10  31.5 
12*  44.9 
7  39.0 

4299 

3382 

4358 

3753 

7429 

3846 

3114 

6845 

7145 

7716 

9505 

10560 

20046 

10271 

10786 

i.S272 

(— )10°26.1' 
(— )  9  22.5 
(— )  4  16.4 
(— )  5  51.3 
(_)  4  17.0 
(— )  8  27.0 
(— )  5  11.2 
(— )  4  20.6 
(— )  9  40.2 
(— )  7  17.2 
(— )  6  59.1 
(— )10  55.3 
(_)  8  58.2 
(— )18  16.0 
(— )17  29.5 
(— )15  27.7 
(— )18  07.7 
(— )11  .55.8 

8.07 

8.. 35 

4.26 

6.22 

4.37 

11.55 

5.27 

3.76 

11.35 

10.18 

10.49 

15.31 

15.00 

30.41 

21.31 

20.87 

25.09 

19.75 

807 
960 
1875 
1355 
2069 
1428 
1691 
2203 
1263 
1695 
1835 
1398 
1763 
1535 
1206 
1331 
1318 
1675 

CHAPTEE  12. 

THE  EFFECT  UPON  THE  RANGE  OF  VARIATIONS  IN  THE  OTHER  BALLISTIC 

ELEMENTS,  WHICH  INCLUDES  THE  DATA  GIVEN  IN  COLUMNS  10, 

11,  12  AND  19  OF  THE  RANGE  TABLES. 

New  Symbols  Introduced. 

AX  ....  Variation  in  the  range  in  feet. 
AR  ....  Variation  in  the  range  in  yards. 
A(sin  2</)) .  . .  .Variation  in  the  sine  of  twice  the  angle  of  departure. 

AvA-  ' .  •  Quantity  appearing  in  Table  II,  in  the  Av  column  pertaining  to  A. 

With  figures  before  the  V  it  shows  the  amount  of  variation  in  V 

for  which  used.     (Be  careful  not  to  confuse  this  symbol  with 

AV  or  8V.) 
BV.  .  .  .Variation  in  the  initial  velocity.     (Be  careful  not  to  confuse  this 

symbol  with  Afa  or  A  7.) 
AV .  . .  .Difference  between  V  for  two  successive  tables  in  Table  II  (Ingalls' 

table  as  originally  computed;  not  the  abridged  tables  repro- 
duced for  use  with  this  text  book)  being  either  50  f.  s.  or  100  f.  s. 

(Be  careful  not  to  confuse  this  symbol  with  Ay  a  or  8V.) 
AVw.  . .  .Variation  in  the  initial  velocity  due  to  a  variation  in  weight  of  pro- 
jectile.   Figures  before  the  w  show  amount  of  variation  in  w  in 

pounds. 
AXy. . .  .Variation  in  the  range  in  feet  due  to  a  variation  in  V.  Figures  before 

the  V  show  the  amount  of  variation  in  V  in  foot-seconds. 
ARy ....  Variation  in  the  range  in  yards  due  to  a  variation  in  V.     Figures 

before  the  V  show  the  amount  of  variation  in  V  in  foot-seconds. 
AC ....  Variation  in  the  ballistic  coefficient  in  percentage. 
AXc.  . .  .Variation  in  the  range  in  feet  due  to  a  variation  in  C.     Figures 

before  the  C  show  the  percentage  variation  in  that  quantity. 
'  ARc.  .  .  .Variation  in  the  range  in  yards  due  to  a  variation  in  C.    Figures 

before  the  C  show  the  percentage  variation  in  that  quantity. 
AS ...  .  Variation  in  the  value  of  8  in  percentage. 
AA's  ....  Variation  in  the  range  in  feet  due  to  a  variation  in  8.    Figures  before 

the  8  show  the  percentage  variation  in  that  quantity. 
ARs  ....  Variation  in  the  range  in  jards  due  to  a  variation  in  8.     Figures 

before  the  8  show  the  percentage  variation  in  that  quantity. 
A)v.  .  .  .Variation  in  w  in  pounds. 
AX u,.  ..  .Variation  in  the  range  in  feet  due  to  a  variation  in  w.     Figures 

before  the  iv  show  the  amount  of  variation  in  that  quantity  in 

pounds. 
AXw  ' ' .  .That  part  of  AX^  in  feet  which  is  due  to  the  variation  in  initial 

velocity  resulting  from  Aw. 
AZu,".  . .  .That  part  of  AA'^  in  feet  which  is'due  to  Atu  directly. 
Ai?u,.  . .  .Variation  in  the  range  in  yards  due  to  a  variation  in  w.     Figures 

before  the  iv  show  the  amount  of  variation  in  that  quantity  in 

pounds. 
ARw  . . .  .That  part  of  ARw  in  yards  which  is  due  to  the  variation  in  initial 

velocity  resulting  from  Aw. 


133 


EXTEEIOE  BALLISTICS 


Hangre 
tables. 


ARw"  •  •  •  ■  That  part  of  Ai^^  in  yards  which  is  due  to  Aw  directly. 

H .  . .  .  Change  in  height  of  point  of  impact  on  vertical  screen  in  feet,  due 
to  a  change  of  AR  in  R.  Figures  as  subscripts  to  the  H  show 
the  change  in  range  necessary  to  give  that  value  of  H. 

198.  The  range  tables  are  computed  for  standard  conditions,  but  there  are 
certain  elements  that  are  not  always  standard;  for  instance,  the  density  of  the 
atmosphere,  which  rarely  is  standard.  The  principal  elements  that  may  vary  from 
standard  are  the  initial  velocity,  variation  in  which  may  result  from  a  variation  of 
the  temperature  of  the  powder  charge  from  standard  or  other  causes;  the  density 
of  the  atmosphere  and  the  weight  of  the  projectile.  In  order  that  a  satisfactory  use 
may  be  made  of  the  range  tables,  it  is  therefore  necessary  to  include  in  them  data 
showing  the  effect  upon  the  trajectory  of  small  variations  from  standard  in  the  ele- 
ments enumerated  above.  Columns  10,  11  and  12  of  the  range  table  therefore  con- 
tain data  showing  the  effect  upon  the  range  of  small  variations  from  standard  in  the 
initial  velocity,  in  the  weight  of  the  projectile  and  in  the  density  of  the  atmosphere, 
respectively.  It  is  the  province  of  this  chapter  to  show  how  the  data  in  these  columns 
is  derived,  and  also  that  in  Column  19,  which  shows  the  effect  upon  the  position  of 
the  point  of  impact  in  the  vertical  plane  through  the  target  of  a  small  variation  of  the 
setting  of  the  "sight  in  range,  or,  as  it  was  formerly  called,  the  sight  bar  height. 

199.  Let  us  take  the  two  principal  equations  of  exterior  ballistics,  namely : 

X=C(Su-Sv)  (178) 


sm2cf>  =  c(\ 


^Su  —  Sv  J 

These  involve  the  range,  X;  the  angle  of  departure,  ^;  the  ballistic  coefficient,  C; 
and  the  initial  velocity,  V;  either  directly  or  through  their  functions  as  given  in 
Table  I.  These  four  are  the  elements  in  which  variations  from  standard  may  be 
expected,  as  indicated  in  the  preceding  paragraph ;  as  a  change  in  the  density  of  the 
atmosphere  involves  a  corresponding  change  in  the  value  of  C,  and  as  a  change  in  the 
weight  of  the  projectile  involves  a  corresponding  change  in  both  the  initial  velocity 
and  ballistic  coefficient,  as  will  be  explained  later. 

200.  For  our  present  purpose,  therefore,  we  wish,  if  practicable,  to  derive  from 
(178)  and  (179)  a  single  differential  equation  in  which  all  four  of  the  quantities 
enumerated  shall  appear  as  variables.  By  a  noteworthy  series  of  mathematical  com- 
binations and  differentiations,  which  need  not  be  followed  here.  Colonel  Ingalls  has 
derived  such  an  equation,  his  result  being 

A{sm  2<f>)  =  -CAvA-{B- A)  AC  +  BC  ^  (180) 

In  this  equation  the  symbol  A  indicates  a  comparatively  small  difference  in  value  or 
differential  increment  (either  positive  or  negative)  in  the  value  of  the  quantity  to 
which  prefixed.  C  is  the  ballistic  coefficient;  <^  is  the  angle  of  departure;  A  and  B 
are  Ingalls'  secondary  functions  as  they  appear  in  Table  II ;  and  A^a  is  the  quantity 
contained  in  Table  II  in  the  A^  column  pertaining  to  A,  where  Afa  is  for  ±  50  or 
It  100  f .  s.  according  to  the  table  used.  For  100  f .  s.  difference  in  velocity  between 
successive  tables,  the  solution  of  the  above  equation  would  give  the  proper  result  as 
it  stands ;  but  for  50  f .  s.  difference  in  velocity,  fifty  one-hundredths,  or  one-half,  of 
the  variation  should  be  taken,  as  shown  later.  Great  care  must  be  exercised  not  to 
confuse  the  three  quantities  represented  by  the  symbols  Afa  as  given  above,  SV,  which 
represents  a  differential  increment  of  the  initial  velocity  and  A7,  which  represents 
the  difference  in  velocity  between  two  successive  tables  in  Table  II.     (Ingalls'  tables 


PEACTICAL  METHODS  133 

as  originally  computed ;  not  the  abridged  tables  reprinted  for  use  with  this  text  book. 
The  values  of  V  given  at  the  top  of  each  table  show  the  value  of  AF  for  that  table.)* 

201.  Having  the  above  general  differential  equation   (180)   involving  all  four  fn'inltfii'"*"* 
variables  with  which  we  wish  to  deal,  we  now  wish  to  apply  it  to  the  several  cases  at   velocity, 
issue;  and  we  will  first  consider  the  variation  in  range  resulting  from  a  small  varia- 
tion in  the  initial  velocity ;  in  other  words,  we  will  find  out  how  to  compute  the  data 
contained  in  Column  10  of  the  range  table.     For  this  case  the  initial  velocity  and 

the  range  are  the  only  variables,  and  AC  and  A  (sin  2^)  therefore  become  zero. 
Equation  (180)  then  becomes 

AXv=^X  (181) 

X) 

in  which  AAV  represents  the  change  in  range  in  feet  resulting  from  a  change  of  SV  in 
initial  velocity.    To  use  this  formula  we  must  first  compute  the  value  of  Z  =  -^f  ,  and, 

then,  from  Table  II,  with  V  and  Z  as  arguments,  we  may  take  the  value  of  B.f 

202.  Suppose  that  we  desire  to  compute  the  change  in  range  at  10,000  yards 
resulting  from  a  variation  in  the  initial  velocity  of  ±50  foot-seconds  from  the 
standard,  for  our  standard  problem  13"  gun  (7  =  2900  f.  s.,  w  =  870  pounds,  c=0.61). 
In  paragraph  157  we  found  that  for  this  problem  Z  =  2984.1  and  log  (7  =  1.00231. 
We  desire  our  result  in  yards,  and  as  there  is  no  quantity  appearing  in  the  equation  in 

*  In  determining  the  value  of  A  y^  from  the  table  for  use  in  the  formula,  the  tabular 
value  must  be  corrected  by  interpolation  for  the  exact  values  of  Z  and  V,  as  follows: 

(a)  Suppose  we  had  V  =  2800  f.  s.  and  Z  =  3773.2.  The  next  lower  tabular  value  of 
AvA  is  .00149,  and  the  difference  between  this  and  the  next  tabular  value  above  it  in  value 
is  .00155  —  .00149  =  .00006.     Therefore  our  value  for  use  in  the  formula  would  be 

^,,  =  . 00149 +  :^M6^^^  ^.0015339 

which  is  carried  out  for  the  full  limit  of  use  with  our  log  tables. 

(b)  Suppose  we  had  V  =  2750  f.  s.  and  Z  =  3770.5.    The  next  lower  tabular  value  of 

70  5 
Afa  is  .00167,  and  as  in  (a)  the  correction  for  Z  would  be  (.00173  —  .00167)  -~  .    The  next 

lower  tabular  value  of  A^^,  as  given  above,  is  .00167,  which  is  for  Z  =  3700;  and  turning 
to  the  table  for  V  =  2800  f.  s..  we  see  that  the  value  of  A^  for  Z  —  3700  f.  s.  is  .00149. 
Therefore  the  variation  in  A,-^  for  100  f.  s.  increase  in  V  would  be  .00149  —  .00167 
=  —  .00018,  and  for  50  f.  s.  it  would  be  half  that.  Our  complete  interpolation  for  this 
case  would  therefore  be 

A ,.  =  .00167  -f  ^^-XIM  _  ,000^X10  ^  _^,^,223 

(c)  In  the  case  of  the  5"  gun  for  which  V  =  3150  f.  s.,  this  interpolation  is  further 
complicated  by  the  fact  that  we  have  no  table  from  which  to  determine  the  value  of  A  y^ 
for  y  =  3200  f.  s.  The  rate  of  change  of  A^^  at  this  point  for  an  increase  of  100  f.  s.  in 
initial  velocity  may  be  obtained  with  sufficient  accuracy  for  every  ordinary  purpose  as 
follows: 

Suppose  y  =  3150  f.  s.  and  Z  —  3770.5 
For  Y  =  2900  f.  s.  and  Z  =  3700  we  have  Ar^  =  .00137 
For  Y  =  3100  f.  s.  and  Z  =  3700  we  have  Ar^  =  .00111 

Therefore,  for  a  change  of  Y  of  200  f.  s.  we  here  have  a  change  in  Arx  of  .00111  —  .00137 
=  —  .00026.  Assuming  that  the  same  rate  of  change  continues  for  the  next  100  f.  s.  increase 
in  V,  which  assumption  is  not  greatly  in  error,  we  would  have  that  the  change  in  the  value 
of  Ar4  between  Y  =  3100  f.  s.  and  y  =  3200  f.  s.  would  be  H— -00026)  =  —  .00013.  Our 
interpolation  would  therefore  become 

A,.,  =  .00111  +  :0000^X70,5  _  -00013^X50  ^  ^^^^^^3 

f  The  convention  employed  in  this  chapter  relative  to  the  double  sign  (±)  is  that  a 
positive  sign  in  a  result  means  an  increase  in  range  and  a  negative  sign  a  decrease. 


134  EXTEEIOE  BALLISTICS 

feet,  we  may  use  the  range  in  yards,  which  will  give  a  result  also  in  yards.  Also  the 
difference  Ay  a  at  this  point  in  the  table  is  for  a  difference  AT''  in  velocity  between  two 
successive  tables   (A7=100  f.  s.,  between  tables  2900  to  3000  f.  s.,  and  3000  to 

3100  f.  s.),  therefore  we  apply  a  factor  ot —y  l^r^  ^'^  this  casej.     Therefore  if 

we  let  ARv  represent  the  change  in  range  in  yards  for  a  variation  of  8T^  =  50  f.  s.  in 
the  initial  velocity,  the  expression  becomes 

ARy=^X^XB 

The  work  then  becomes,  from  Table  II : 

ArA  =  . 00099+  '^^^^/qq  ^^'"^  =.001024 

5=.0178+  -^^^^^^^-^  =.01856 

ArA  =  .001024    log  7.01030-10 

87=  ±50    ±log  1.69897 

i?=  10000    log  4.00000 

£  =  .01856    log  8.26857-10 colog  1.73142 

A7  =  100    log  2.00000 colog  8.00000-10 

ARv=  ±276  yards ±iog  2.44069 

and  the  signs  show  that  an  increase  in  initial  velocity  will  give  an  increase  in  range, 

and  the  reverse,  which  was  of  course  to  be  expected. 
For  variation  203.  Again,  suppose  that  the  density  of  the  air  varies  from  standard,  as  it 

^"atmosphere,    generally  does,  and  we  wish  to  determine  the  resultant  effect  upon  the  range.     We 

know  that  C  =  J-^n: ,  and  in  this  case  the  only  variables  are  X  and  8,  as  <i,  V  and  w 

are  supposed  to  be  constant.  A  change  in  the  value  of  8  therefore  causes  a  change  of 
the  same  amount  in  the  value  of  C,  but  as  8  appears  in  the  denominator  of  C,  an 
increase  of  a  certain  per  cent  in  the  value  of  8  will  cause  a  decrease  of  the  same  per 
cent  in  the  value  of  C,  that  is,  a  ±AS  gives  a  tAC  Equation  (180)  therefore 
becomes 

BC  ^^-^=-{B-A)AG 

^..           (B-A)X  ^  AC  f.oo\ 

or  AAc=-^ p  ^      X  -g-  (182) 

204.  As  an  example  of  the  use  of  this  formula,  let  us  take  the  same  data  as  in 
paragraph  202,  and  compute  the  change  of  range  resulting  from  a  variation  from 
standard  of  ±  10  per  cent  in  the  density  of  the  atmosphere,  letting  AR-^qS  represent  the 
desired  result  in  yards,  and  again  substituting  R  for  X  to  get  the  result  in  yards. 
Equation  (182)  then  becomes 

£  =  .018560 
A  =  .014601 


B-A  =  .00395d log  7.59759-10 

£  =  10000    log  4.00000 

£=.01856    log  8.26857-10 colog  1.73143-10 

4^  =  ±.l    ±log  9.00000-10 

8 

A£io5=  +313  yards ±log  2.32903 


PRACTICAL  METHODS 


135 


Carrying  through  the  signs  shows  that  an  increase  in  the  density  gives  a  decrease  in 
the  range,  and  the  reverse,  which  was  to  be  expected. 

205.  Again,  suppose  it  is  the  weight  of  the  projectile  that  varies,  the  other  ele-  Effect  of 
ments  remaining  fixed,  and  we  wish  to  determine  the  resultant  change  in  the  range,  weight  of^ 
This  change  is  composed  of  two  parts.  As  the  charge  which  furnishes  the  propelling  '^^°^^^ 
power  for  the  projectile  is  supposed  to  remain  fixed,  a  heavier  projectile  will  leave 
the  gun  with  a  less  initial  velocity  than  a  lighter  one,  which  would  result  in  a  decrease 
in  range,  as  already  seen.  The  second  part  is  the  result  of  the  change  in  weight 
affecting  the  flight  after  leaving  the  gun;  and  it  will  be  seen  that,  of  two  shell  of 
different  weights  leaving  the  gun  with  the  same  initial  velocity,  the  heavier  will  have 
the  greater  momentum  and  therefore  the  greater  range.  The  two  effects  are  there- 
fore of  opposite  sign,  the  heavier  shell  tending  first  to  reduce  the  initial  velocity  with 
which  the  shell  leaves  the  gun,  but  after  so  leaving  tending  to  increase  the  range, 
through  its  greater  momentum,  over  what  would  have  been  the  range  of  a  standard 
weight  projectile  leaving  the  gun  with  the  same  reduced  initial  velocity.  This  second 
part  of  the  variation  is  represented  by  a  change  in  the  value  of  the  ballistic  coeificient, 
again,  as  was  the  case  for  a  variation  in  atmospheric  density,  of  the  same  per  cent 
value  as  the  per  cent  variation  in  the  weight  of  the  projectile,  but  this  time  with  the 
same  sign,  as  w  appears  in  the  numerator  of  the  expression  for  the  value  of  the 
ballistic  coefficient.  A  change  of  ±Aiv  will  therefore  give  a  corresponding  change 
of  ±AC. 

206.  Let  us  now  consider  first  the  change  in  range  due  to  the  variation  in  the   For  variation 
initial  velocity  resulting  from  the  variation  in  the  weight  of  the  projectile.     By  a    proTe^cuie.  ° 
formula  taken  from  interior  ballistics,  the  derivation  of  which  needs  no  inquiry  here, 
we  have 


8V=-M—  V 
w 


(183) 


in  which  .¥  =  0.36  for  guns  C,  F,  H,  J,  K,  M,  N,  P,  Q  and  R  Other  values  of  M 
were  used  in  computing  the  range  tables  for  other  guns  and  projectiles  given  in  the 
edition  of  the  Eange  and  Ballistic  Tables  published  for  use  with  this  text  book,  but 
work  under  this  head  will  here  be  confined  to  the  guns  enumerated  above  (for 
il/z=0.36),  and  no  inquiry  into  other  values  of  M  is  necessary  here. 
And  from  (180) 


(184) 


but  Afa  is  the  difference  for  AF  at  that  part  of  the  table,  and  we  must  therefore 

87 
mtroduce  the  factor  — ^  ;  and  we  may  also  substitute  R  for  X  throughout,  which 

gives  us 

^RJ=^X^  XR  (185) 


in  which  Afa  in  (184)  and  (185)  is  the  quantity  from  Table  II  corresponding  to  the 
given  value  of  Z. 

207.  ISTow  for  the  second  part  of  this  change,  that  due  to  the  variation  in 
momentum  resulting  from  the  variation  in  weight,  but  acting  only  after  the  pro- 
jectile has  been  expelled  from  the  gun  at  the  reduced  initial  velocity  determined 
above,  which  is  the  part  that  affects  the  value  of  C.  From  what  has  already  been 
explained  we  readily  have  for  this 

A(7   _  {B-A)X  ..  Am; 
C    ~        B 


AZe=  (^-/)^  X 


X 


(186) 


136  EXTEEIOR  BALLISTICS 

and  substituting  R  for  X  to  get  the  result  in  yards 

^RJ'  =  i^^^X^  (187) 

To  combine  the  two  results  to  get  the  total  change  in  range  resulting  from  both 
causes  in  yards,  we  would  have 

^R^  =  ^EJ  +  ^RJ'=  %^  X  -^  XR+  i^-^)R  X  ^  (188) 

the  sign  of  the  first  term  of  the  second  member  being  inverted  to  make  the  last  two 
terms  of  opposite  sign  and  the  final  result  of  the  proper  sign. 

208.  As  an  illustration  of  the  use  of  this  formula,  let  us  revert  once  more  to 
our  standard  problem  as  given  in  paragraph  203,  and  compute  the  change  in  range 
in  that  case  resulting  from  a  variation  from  standard  of  ±10  pounds  in  the  weight 
of  the  projectile.    Using  the  formulae  given  in  paragraphs  206  and  207  we  have 
Afa  =  . 001024         J.  =  .014G01         5=. 018560 

Aw=±10    ±log  1.00000 

w  =  870    log  2.93952 colog  7.06048-10 

7  =  2900    log  3.46240 

ilf=.36    log  9.55630-10 

87=  +12  f.  s ±log  1.07918 

Af4  =  .001024    log  7.01030-10 

87=  =P  12    ^log  1.07918 

22  =  10000    log  4.00000 

5  =  . 01856    log  8.26857 -10.. colog  1.73143 

A7  =  100    log  2.00000 colog  8.00000-10 

Ai?^'= +66.21   =plog  1.82091 

£  =  .018560 
A  =  .014601 


.S-A  =  .003959    log  7.59759-10 

22  =  10000    log  4.00000 

5  =  . 01856   log  8.26857-10.  .colog  1.73143 

Aw=  ±10   ±log  1.00000 

m;  =  870    log  2.93952 colog  7.06048-10 

LRJ'=  ±  24.52    ±  log  1.38950 

Ai^TO^  =1=41.62  yards 

which  shows  that  for  this  gun,  at  this  range,  an  increase  of  10  pounds  above  standard 

in  the  weight  of  the  projectile  decreases  the  range  41.6  yards,  and  the  reverse.    Note 

also  that  here  a  positive  value  of  Aw  gives  a  negative  value  of  Ai^^,  but  that  in  the 

range  table  there  is  no  negative  sign  attached  to  the  figures  in  the  appropriate  column. 

The  above  is  of  course  the  correct  mathematical  convention,  but  after  the  work  is  all 

done,  as  a  decrease  in  range  is  the  normal  and  general  result  of  an  increase  in  weight, 

in  making  up  the  range  tables  such  a  decrease  is  considered  as  positive  and  the  signs 

in  the  tables  are  given  accordingly. 

Short  method       209.  The  above  is  the  general  method,  but  in  actually  computing  the  data  for 

^?n  JefgM  of   the  range  tables  there  is  a  short  cut  that  may  advantageously  be  used  to  reduce  the 

projectue.    jj^-j^Q^jit  of  labor  involved  in  the  computations  for  Column  11.    If  we  first  compute 

the  data  for  Columns  10  and  12,  as  is  actually  done  in  such  computations  and  as  we 

have  already  done  here;  that  is,  if  we  have  already  found,  for  the  given  range,  the 


PEACTICAL  METHODS  137 

chancre  in  range  resulting  from  a  variation  in  the  initial  velocity  of  ±8V,  and  also 
that  resulting  from  a  variation  of  ±  AC  in  the  value  of  the  ballistic  coefficient  which 
is  the  same  as  that  due  to  a  variation  of  q^^AS  in  the  density  of  the  air,  we  readily 
derive  the  following  formulae : 

AR^'  =  ARvX  jyr  (189) 

ARJ'  =  AR5X  -^  (190) 

ARy,  =  ARJ  +  AR^o"  =  ARvX   1^   +AR5X    -^    XAS  (191) 

0  y  w 

the  two  terms  being  combined  with  the  proper  signs. 

For  our  given  problem  the  work  then  becomes,  after  finding  that  V  =  12  f.  s.  in 
the  same  way  that  we  did  before 

Ai2y  =  376    log  2.44070 

87=  :;:  12    ^log  1.07918 

87' ==50    log  1.69897 colog  8.30103-10 

Ai^^o'=  +66.21    +log  1.82091 

Ai?5  =  213    log  2.32901 

Aiu=±10    ±log  1.00000 

w  =  870    log  2.93952 colog  7.06048-10 

A8  =  10    log  1.00000 

Ai2''=  ±24.52    ±log  1.38949 


Ai2,,;= -41.62  yards 

Note  that  the  logarithms  used  above  for  276  and  213  are  not  taken  from  the 
log  table,  but  are  the  exact  logarithms  resulting  from  the  previous  work,  as  given  in 
paragraphs  202  and  204. 

The  AS  =  10  in  the  last  part  of  the  above  work  comes  in  becaase  the  Ai25  =  213 
is  for  10  per  cent  variation  in  density ;  therefore  for  100  per  cent  variation  it  would 

be  ASxAiS5  =  313x10,  of  which  we  take  — •  =  77^^. 

w         870 

210.  We  will  now  investigate  the  method  of  computing  the  data  contained  in   change  of 

Column  19  of  the  range  tables ;  that  is,  of  determining  how  much  vertical  displace-  pact  in  verti- 

ment  in  the  vertical  plane  through  the  target  at  the  given  range  will  result  from  an 

increase  or  decrease  of  a  few  yards  in  the  setting  of  the  sight  in  range ;  or,  as  it  was 

formerly  called,  in  the  sight  bar  height. 


cal  plane. 


138 


EXTERIOE  BALLISTICS 


211.  Assuming  that,  for  flat  trajectories,  when  the  point  of  fall  is  not  far  from 
the  target  as  compared  to  the  range,  the  portion  of  the  trajectory  between  the  target 
and  the  point  of  fall  is  practically  a  straight  line,  we  see  from  Figure  15,  in  which 
AB^^X,  AC  =  H  and  ABC  =  w,  that,  if  we  let  H  represent  the  vertical  change  of 


FiGUEE   15. 


point  of  impact  in  feet,  and  AX  the  change  in  range  that  will  correspond,  also  in  feet, 
we  will  have 


tanw=  -^ ,  or  H  =  AXtan^ 


(193) 


212.  As  an  example,  we  will  take  our  standard  problem,  for  a  range  of  10,000 
yards,  and  will  find  the  change  in  the  point  of  impact  in  the  vertical  plane  through 
the  target  resulting  from  a  change  in  sight  setting  of  dzlOO  yards.  By  equation 
(192)  the  work  becomes 

AZ=  ±300    ±log  2.47712 

(0  =  5°  21'  11"    tan  8.97174-10 

ffioo^^SS  feet ±log  1.44886 

The  value  of  w  employed  above  is  taken  from  paragraph  157  of  Chapter  8,  where  we 
explained  the  opening  work  of  computation  relative  to  this  particular  trajectory. 


EXAMPLES. 

1.  Require  to  be  taken  from  the  range  tables  the  amount  of  change  of  range 
resulting  from  any  reasonable : 

(a)  Variation  from  standard  in  the  initial  velocity. 

(b)  Variation  from  standard  in  the  weight  of  the  projectile. 

(c)  Variation  from  standard  in  the  density  of  the  atmosphere.  The  readings 
of  barometer  and  thermometer  should  be  given,  and  determination  of  change  of  range 
made  by  use  of  Table  IV,  Also  exercise  in  the  same  problem,  using  Table  III  instead 
of  Table  IV. 

(d)  In  addition  to  the  above,  call  for  the  taking  from  the  tables  of  the  effect 
upon  the  range  of  two  or  more  of  the  above  variations  combined. 

(e)  Also  call  for  the  determination  from  the  tables  of  the  change  of  the  point  of 
impact  in  the  vertical  plane  resulting  from  a  change  in  the  setting  of  the  sight  in 
range,  and  vice  versa. 


PEACTICAL  METHODS 


139 


2.  Compute  the  change  in  range  resulting  from  the  variation  from  standard  in 
the  initial  velocity  given  below,  all  other  conditions  being  standard. 


DATA. 

ANSWERS. 

Problem. 

Projectile. 

Velocity, 
f.  s. 

Range. 

Yds. 

Maxi- 
mum, 
ordinate. 
Feet. 

Variation 

in  initial 

velocity. 

f.  s. 

Change  in 

d. 
In. 

w. 
Lbs. 

c. 

range. 
Yds. 

A 

3 
3 
4 
5 

5 
6 
6 
6 
7 
7 
8 

10 
10 
12 
13 
13 
14 
14 

13 

13 

33 

50 

50 

105 

105 

105 

165 

165 

260 

510 

510 

870 

11.30 

1130 

1400 

1400 

1.00 
1.00 
0.67 
1.00 
0.61 
0.61 
1.00 
0.61 
1.00 
0.61 
0.61 
1.00 
0.61 
0.61 
1.00 
0.74 
0.70 
0.70 

11.50 
2700 
2900 
3150 
3150 
2600 
2800 
2800 
2700 
2700 
2750 
2700 
2700 
2900 
2000 
2000 
2000 
2600 

2000 

3600 

3000 

4000 

4000 

11.500 

4500 

3500 

7000 

7000 

8000 

10000 

11000 

21000 

10000 

11000 

14000 

14000 

1.50 

180 

55 

125 

90 

1887 

169 

75 

563 

395 

485 

1085 

997 

4218 

1701 

1830 

3204 

1790 

+25 
—30 
+.50 
—60 
+75 
—45 
+50 
—40 
+40 
—60 
+60 
—45 
+45 
—70 
+70 
—60 
+45 
—45 

+  49.8 

B 

—  44.7 

c 

+  83.4 

D 

—  98.4 

E 

+  144.8 

F 

—226.8 

G 

+  110.5 

H 

—  82.9 

I 

+127.4 

,J 

—227.9 

K 

+260.7 

L 

—211.2 

M  ..             

+266.9 

N" 

—647.2 

0 

+460.7 

p 

—467.3 

0 

+431.8 

R 

—347.6 

Note. — The  above  results  vary  slightly  from  those  taken  from  the  range  tables.  The 
reason  for  this  is  that  the  range  table  results  are  more  accurately  determined  by  using 
the  corrected  value  of  C,  found  by  the  methods  of  Chapter  8,  Example  7;  whereas  in  the 
above  the  value  of  C  is  determined  by  the  use  of  a  value  of  /  determined  by  the  use  of  the 
maximum  ordinate  given  above  and  of  Table  V.  The  same  variation  from  range  table 
results  will  be  found  in  all  problems  in  which  this  latter  process  is  employed. 


3.  Compute  the  change  in  range  resulting  from  the  variation  from  standard  in 
the  density  of  the  atmosphere  given  below,  other  conditions  being  standard. 


DATA. 

ANSWERS. 

Problem. 

Projectile. 

Velocity. 

f.  s. 

Range. 

Yds. 

Maxi 

mum 

ordinate. 

Feet. 

Varia- 
tion in 
densitv 

Change  in 

d. 
In. 

Lbs 

c. 

range. 
Yds. 

A 

3 

3 

4 

5 

5 

6 

6 

6 

7 

7 

8 

10 

10 

12 

13 

13 

14 

14 

13 

13 

33 

50 

50 

105 

105 

105 

165 

165 

260 

510 

510 

870 

1130 

1130 

1400 

1400 

1.00 
1.00 
0.67 
1.00 
0.61 
0.61 
1.00 
0.61 
1.00 
0.61 
0.61 
1.00 
0.61 
0.61 
1.00 
0.74 
0.70 
0.70 

1150 

2700 
2900 
31.50 
3150 
2600 
2800 
2800 
2700 
2700 
27.50 
2700 
2700 
2900 
2000 
2000 
2000 
2600 

1600 

3000 

3500 

3000 

3000 

13000 

3000 

2500 

6000 

6000 

7000 

9000 

10000 

22000 

9000 

10000 

13000 

13000 

79 

104 

80 

57 

45 

2725 

61 

35 

363 

269 

3,50 

807 

784 

4801 

1292 

1438 

2637 

1480 

+  5 

+10 
—10 

+  7 

—  5 
+  6 

—  6 

+  8 

—  8 
+  12 
—12 

+  7 

—  7 
+  6 

—  6 

+  5 

—  5 

—  10.9 

B 

+  85.7 

C 

—  83.7 

D 

+  90.9 

E 

—  40.5 

F 

+298.3 

G 

—  39.7 

H 

+  18.2 

I 

—173.1 

J 

+  113.3 

K 

—190.3 

L 

+384.3 

M 

—177.8 

N 

+617.5 

0 

—1.55.7 

P 

+  147.2 

Q 

—176.1 

R 

+  175.1 

140 


EXTERIOE  BALLISTICS 


4.  Compute  the  change  in  range  resulting  from  the  variation  from  standard  in 
the  weight  of  the  projectile  given  below,  other  conditions  being  standard.  Use  the 
direct  method,  without  the  use  of  Columns  10  and  12  of  the  range  tables. 


DATA. 

ANSWERS. 

Problem. 

Projectile. 

Velocity, 
f.  s. 

Range. 
Yds. 

Maxi- 
mum 
ordinate. 
Feet. 

Varia- 
tion in 
weight. 
Lbs. 

Change  in 

d. 
In. 

Lbs. 

c. 

range. 
Yds. 

C 

4 
6 
6 
7 
8 

10 
12 
13 
14 
14 

33 

105 

105 

165 

260 

510 

870 

1130 

1400 

1400 

0.67 
0.61 
0.61 
0.61 
0.61 
0.61 
0.61 
0.74 
0.70 
0.70 

2900 
2600 
2S00 
2700 
2750 
2700 
2900 
2000 
2000 
2600 

3000 

13600 

2600 

6100 

7100 

10100 

21000 

10100 

13100 

13100 

55 
3123 

38 

280 

362 

804 

4218 

1474 

2690 

1509 

+  1 

—  2 

—  5 

—  3 
+  4 
+  5 

—  5 
—15 
+  12 
—12 

—33.7 

F 

— 22  2 

H 

+62.0 

J 

+34.1 

K 

M 

—.35.4 

—27.7 

N 

+  8.3 

P 

+36.7 

0 

—26.0 

R 

+28.4 

Note. — The  signs  in  the  above  results  are  mathematically  correct;  that  is,  a  positive 
result  means  an  increase  in  range.  But  remember  the  convention  reversing  these  signs 
in  the  range  tables,  whereby  a  positive  sign  (or  no  sign)  means  a  decrease  in  range,  the 
normal  and  most  common  result  of  an  increase  in  weight  of  projectile. 


5.  Compute  the  change  in  range  resulting  from  the  variation  from  standard  in 
the  weight  of  the  projectile  given  below,  other  conditions  being  standard;  using  the 
data  from  Columns  10  and  12  of  the  range  tables. 


DATA. 

ANSWERS. 

Problem. 

Projectile. 

r. 

f.    9. 

Range. 
Yds. 

Varia- 
tion in 
weight. 
Lbs. 

Col.  10. 

Change   in 

range    for 

var.  of    -f-50 

f.  8.  in  I.  V. 

Yds. 

Col.  12 

Change   in 

range  for 

var.of  ±10c^o 

in  density. 

Yds. 

Change  in 

Range. 

Yds. 

d. 
In. 

Lbs. 

c. 

C 

4 

33 

0.67 

2900 

4000 

+  1 

103 

109 

—32.1 

F 

6 

105 

0.61 

2600 

12000 

—  2 

260 

531 

—  8.4 

H 

6 

105 

0.61 

2800 

3500 

0 

103 

57 

+71.7 

J 

7 

165 

0.61 

2700 

7000 

—  5 

190 

188 

+55.0 

K 

8 

260 

0.61 

2750 

8000 

+  7 

217 

204 

—60.8 

M 

10 

510 

0.61 

2700 

11000 

+  8 

296 

300 

—43.2 

N 

12 

870 

0.61 

2900 

18500 

—  8 

432 

648          1 

+23.4 

P 

13 

1130 

0.74 

2000 

11000 

—12 

391 

286         ! 

+29.4 

Q 

14 

1400 

0.70 

2000 

14000 

+  12 

484 

392 

—26.1 

R 

14 

1400 

0.70 

2600 

14000 

—13 

388 

395         1 

1 

+30.8 

Note. — See  note  to  Example  4  about  signs,  which  applies  to  this  example  also. 


PEACTICAL  METHODS 


141 


6.  Compute  the  change  in  the  position  of  the  point  of  impact  in  the  vertical 
plane  through  the  target  for  the  following  variations  in  the  setting  of  the  sight  in 
range,  taking  the  values  of  the  angle  of  fall  from  the  range  tables.  Conditions 
standard. 


DATA. 

ANSWERS. 

Projectile. 

Velocity. 

f.    3. 

Range. 
Yds. 

1 

Variation  in 

setting    of 

sight  in 

range. 

Yds. 

+  =:incr'se. 

—  =:decr'se.! 

Change   in 

point   of 

impact. 

Ft. 

+  =  raise. 

—  =L  lower. 

d. 
In. 

W. 
Lbs. 

c. 

A 

3 
3 

4 
5 
5 
6 
6 
6 
7 
7 
8 

10 
10 
12 
13 
13 
14 
14 

13 

13 

33 

50 

50 

105 

105 

105 

165 

165 

260 

510 

510 

870 

1130 

1130 

1400 

1400 

1.00 
1.00 
0.67 
1.00 
0.61 
0.61 
1.00 
0.61 
1.00 
0.61 
0.61 
1.00 
0.61 
0.61 
1.00 
0.74 
0.70 
0.70 

1150 
2700 
2900 
3150 
3150 
2600 
2800 
2800 
2700 
2700 
2750 
2700 
2700 
2900 
2000 
2000 
2000 
2600 

2100 

3600 

4000 

3600 

3600 

14000 

3600 

3100 

6600 

6600 

7600 

9600 

10600 

17000 

9600 

10600 

13600 

13600 

+  50 

—  50 
+100 
—100 
+  75 

—  75 
+  60 

—  60 
+  80 

—  80 
+  90 

—  90 
+110 
—110 
+  70 

—  70 
+  60 

—  60 

+  17.4 

B 

—13.5 

c 

+  13.2 

D 

—12.9 

E 

+  6.6 

F 

—98.4 

G 

+  7.4 

H 

—  4.7 

I 

+29.5 

J 

—19.2 

K 

+23.2 

L 

—45.9 

M 

+44.0 

N 

—75.5 

0 

+.53.1 

P 

—51.0 

0 

+61.4 

R 

—34.6 

PART  III. 

THE  VARIATION  OF  THE  TRAJECTORY  FROM  A 

PLANE  CURVE. 

INTEODUCTION  TO  PART  III. 

Having  completed  the  consideration  of  the  general  trajectory  in  air  for  all 
velocities  when  considered  as  a  plane  curve,  by  means  of  the  differential  equations 
connected  therewith,  and  having  seen  that  for  the  purposes  discussed  in  Parts  I 
and  II  no  material  error  is  introduced  into  the  results  by  such  assumption  that  the 
trajectory  is  a  plane  curve,,  we  now  come  to  the  question  of  how  to  hit  a  given  spot 
with  the  projectile  from  a  given  gun,  under  given  conditions,  so  far  as  the  deflection 
of  the  projectile  from  the  original  plane  of  fire  is  concerned.  Although  such  varia- 
tion will  not  materially  affect  the  results  of  computations  of  the  values  of  the  ranges, 
angles,  velocities,  times,  etc.,  as  already  shown,  it  will  at  once  be  apparent  that  a 
variation  of  a  very  few  yards  from  the  original  plane  of  fire  will  cause  a  miss,  unless 
compensated  for  in  the  sighting  of  the  gun.  In  Part  III  we  therefore  take  up  the 
study  of  the  forces  acting  to  deflect  the  projectile  from  the  original  plane  of  fire, 
thereby  causing  a  miss  unless  compensated.  These  are  drift,  wind  and  motion  of  the 
gun  or  target;  and  expressions  will  be  derived  to  determine  the  extent  of  the  deflec- 
tions arising  from  each  cause,  and  methods  will  be  devised  for  applying  the  necessary 
corrections  in  aiming  to  overcome  these  errors. 


CHAPTEE  13. 

DRIFT  AND  THE  THEORY  OF  SIGHTS,  INCLUDING  THE  COMPUTATION  OF 
THE  DATA  CONTAINED  IN  COLUMN  6  OF  THE  RANGE  TABLES. 

New  Symbols  Introduced. 

/Li= -pj  ....In  which  Ic  is  the  radius  of  gyration  of  the  projectile  about  its 
longitudinal  axis,  and  R  is  the  radius  of  the  projectile. 
-y-  . . .  .A  special  ratio  explained  in  the  text. 

n.  . .  .Twist  of  the  rifling  at  the  muzzle. 
D' .  . .  .  Ingalls'  secondary  function  for  drift. 
D .  . .  .Drift  in  yards. 

I.  . .  .Sight  radius. 

i.  . .  .  Permanent  angle. 
h.  . .  . Sight  bar  height  in  inches. 
D.  . .  .Deflection  in  yards  (used  with  R  in  yards). 

213.  Experience  shows  that  the  projectile  from  any  rifled  gun,  when  fired  in   Drift, 
still  air,  deviates  from  the  plane  of  fire  (a  vertical  plane  through  the  axis  of  the  gun) 

to  an  extent  approximately  proportional  to  the  square  of  the  time  of  flight,  and  in  the 
direction  towards  which  the  upper  surface  of  the  projectile  moves  in  its  rotation. 
This  deviation  is  called  the  "  drift,"  and  for  all  United  States  naval  guns  is  to  the 
right,  since  these  guns  are  so  rifled  that  their  projectiles,  viewed  from  the  rear,  rotate 
with  the  hands  of  a  watch  when  in  flight. 

214.  Since  the  drift  increases  more  rapidly  in  proportion  than  does  the  range, 
the  horizontal  projection  of  the  trajectory  is  really  a  curve  convex  to  the  horizontal 
trace  of  the  plane  of  fire.  N'evertheless,  though  the  absolute  value  of  the  drift, 
especially  at  long  ranges,  is  too  great  to  be  neglected  in  the  practical  use  of  guns,  its 
relative  value  is  always  small  enough  to  justify  the  assumption  hitherto  made  that 
the  trajectory  is  a  plane  curve,  so  far  as  the  purposes  for  which  we  proceeded  on  that 
assumption  are  concerned.  In  other  words,  the  actual  trajectory  difi'ers  so  little  from 
its  projection  upon  the  plane  of  fire  that  no  appreciable  error  results  from  regarding 
them  as  coincident  in  taking  account  of  the  effect  of  the  resistance  of  the  atmosphere 
upon  the  range,  time  of  flight,  etc.  When  it  comes  to  the  question  of  hitting  a  given 
target,  however,  where  a  very  few  yards  deviation  from  the  original  plane  of  fire  will, 
if  not  compensated,  make  the  difference  between  a  hit  and  a  miss,  the  case  is  far 
different,  and  the  drift  must  therefore  be  taken  into  account  in  discussing  the  sighting 
of  guns. 

215.  The  cause  of  drift  is  that,  soon  after  the  projectile  leaves  the  gun,  the  line    cause  of 
of  action  of  the  air  resistance  ceases  to  coincide  with  .the  axis  of  the  projectile,  on     "  *' 
account  of  the  curvature  of  the  trijectory;  and,  meeting  that  axis  obliquely  between 

the  point  of  the  projectile  and  the  center  of  gravity,  tends  to  raise  the  point;  which 
action,  combined  with  that  of  the  rotation,  causes  the  point  to  move  first  to  one  side 
(to  the  right  for  right-handed  rotation)  and  then  do^vnward.  This  movement,  by 
virtue  of  which  the  axis  of  the  projectile  tends  to  describe  a  cone  about  the  tangent  to 
the  trajectory,  is  called  the  "  precession,"  and  its  result,  in  combination  with  the 
angular  motion  of  the  tangent  caused  by  the  curvature  of  the  trajectory,  is  to  keep 
the  point  of  the  projectile  always  on  that  side  of  the  plane  of  fire  towards  which  it 
10 


146  EXTEEIOE  BALLISTICS 

was  first  deflected.  (The  imprints  of  projectiles  at  their  points  of  fall  upon  the  ground 
at  long  ranges  show  this  to  be  the  case.)  Therefore,  with  right-handed  rifling,  the 
projectile  during  its  flight  always  points  very  slightly  to  the  right  of  the  direction  of 
motion;  and,  as  a  result,  the  resistance  of  the  air  has  a  component  normal  to  that 
direction,  which  carries  the  projectile  bodily  to  the  right  with  increasing  velocity. 
(This  applies  only  to  direct  fire.  When  the  angle  of  departure  exceeds  70°,  as  it 
sometimes  does  in  mortar  fire,  the  drift  is  reversed  in  direction.  The  reason  for  this 
appears  to  be  that,  at  the  vertex  of  the  trajectory,  the  direction  of  the  tangent  changes 
so  suddenly  that  the  slow  movement  of  precession  is  insufficient  to  cause  the  axis  of 
the  projectile  to  keep  pace  with  it.  The  angle  between  the  tangent  and  the  axis  of  the 
projectile  therefore  becomes  greater  than  90°;  the  projectile  moves  approximately 
base  first;  the  resistance  of  the  air  acts  upon  the  opposite  side  of  the  projectile  from 
that  upon  which  it  acted  in  the  ascending  branch  of  the  curve ;  and  the  lateral  move- 
ment to  the  right  is  speedily  checked  and  reversed.  With  these  very  high  angle 
trajectories  the  projectile  always  strikes  base  first.) 
Computation  216.  The  precise  experimental  determination  of  the  amount  of  drift  is  a  matter 

of  drift.  . 

of  great  difficulty,  as  its  value  is  materially  affected  by  lateral  wind  pressure  and  by 
unavoidable  differences  between  different  projectiles.  For  computing  its  value, 
Mayevski  derived  an  approximate  formula,  which  has  been  reduced  by  Ingalls  to  the 
form 

D=JLx~X-^^  (193) 

n        k       cos^  ^ 

In  which 

fx.  =  ^^,  where  Ic  is  the  radius  of  g}Tation  of  the  projectile  and  E  is  its  radius. 

-7-  =a  quantity  which  depends  upon  the  length  of  the  projectile,  the  shape  of 

the  head,  the  angle  which  the  resultant  resistance  makes  with  the 

axis  and  the  distance  of  the  center  of  pressure  from  the  center  of 

gravity. 
n  =  the  twist  of  the  rifling  in  calibers  at  the  muzzle,  that  is,  the  distance  in 

calibers  that  the  projectile  advances  along  the  trajectory  at  the 

muzzle  while  making  one  revolution. 
C  =  the  ballistic  coefficient. 
</>  =  the  angle  of  departure. 
Z^'  =  Ingalls'  secondary  function  for  drift,  to  be  found  in  Table  II,  with  Y  and 

Z  as  arguments. 
Z)  =  drift  in  yards  for  the  given  range  and  angle  of  departure. 


VARIATION  OF  THE  TRAJECTORY  FROM  A  PLANE  CURVE     147 


217.  It  has  also  been  found  necessary  in  some  cases  to  multiply  the  results 
obtained  by  the  use  of  the  above  formula  by  a  certain  empirical  multiplier  in  order 
to  get  correct  results.    The  data  required  for  the  drift  computation  is  therefore : 


DATA. 

Problem. 

Gui 

1  and  projectile. 

Velocity. 

f.  s. 

fi. 

X 

h  ' 

n. 

Multi- 

d. 
In. 

w. 
Lbs. 

c. 

plier. 

A 

3 
3 

4 

5 

5 

6 

6 

6 

7 

7 

8 

10 

10 

12 

13 

13 

14 

14 

13 

13 

33 

50 

50 

105 

105 

105 

165 

165 

260 

510 

510 

870 

11.30 

1130 

1400 

1400 

1.00 
1.00 
0.67 
1.00 
0.61 
0.61 
1.00 
0.61 
1.00 
0.61 
0.61 
1.00 
0.61 
0.61 
1.00 
0.74 
0.70 
0.70 

11.50 
2700 
2000 
3150 
3150 
2600 
2800 
2800 
2700 
2700 
2750 
2700 
2700 
2900 
2000 
2000 
2000 
2600 

0.53 
0.53 
0..53 
0.53 
0.53 
0..53 
0.53 
0.53 
0..53 
0.53 
0.53 
0..53 
0.53 
0.53 
0.53 
0.53 
0.53 
0.53 

0..32 
0..32 
0.32 
0.32 
0.32 
0.32 
0..32 
0.32 
0.32 
0.32 
0..32 
0.32 
0.32 
0.32 
0.32 
0.32 
0.32 
0.32 

25 
25 
25 
25 
25 
25 
25 
25 
25 
25 
25 
25 
25 
25 
25 
25 
25 
25 

1.0 

B 

1.0 

c 

1.0 

D 

1.5 

E 

1.5 

F 

1.5 

G 

1.0 

H 

1.5 

I 

1.0 

J 

1.5 

K 

1.5 

L 

1.0 

M 

1.5 

N 

1.5 

0 

1.0 

p 

1.5 

0 

1.5 

E 

1.5 

218.  Returning  to  our  standard  problem,  we  will  compute  the  drift  for  the 
12"  gun  (7  =  2900  f.  s.,  w  =  870  pounds,  c  =  0.61)  for  a  range  of  10,000  yards,  for 
which  conditions  we  found  in  Chapter  8  that  the  angle  of  departure  was  4°  13'  14", 
Z  =  2984.1  and  log  C  =  1.00231.    The  computation  then  becomes 


1..5 


log  0.17609 


f^ 


=  0.-53    log  9.72428-10 

=  0.32    log  9.50515-10 


/i  =  25    log  1.39794 colog  8.60206-10 

C=    log  1.00231 2  log  2.00462 

Z;'  =  25.6    log  1.40824 

(/)  =  4°   13'  14"    sec  0.00118 3  sec  0.00354 

Z)  =  26.5  yards log  1.42398 

219.  Guns  are  usually,  and  naval  guns  always,  pointed  by  directing  what  is  sights. 
called  the  "  line  of  sight "  at  the  target.  Originally  the  upper  surface  of  the  gun 
itself  was  used  as  the  line  of  sight;  this  was  called  "  sighting  by  the  line  of  metal," 
and  resulted  in  giving  to  the  gun  an  angular  elevation  corresponding  to  the  differ- 
ence in  thickness  of  metal  at  the  breech  and  at  the  muzzle.  Later  on,  a  piece  of  wood, 
called  a  "  dispart,"  was  secured  to  the  muzzle,  so  as  to  give  a  line  from  breech  to 
muzzle  parallel  to  the  axis  of  the  gun.  Such  a  line  of  sight  had  to  be  directed  more 
or  less  above  the  target  according  to  the  range.  Early  in  the  last  century  came  into 
use  the  method  of  having  one  fixed  and  one  movable  sight,  so  that  the  line  between 
them,  which  is  the  line  of  sight,  could  be  adjusted  at  any  desired  angle  with  the  axis 
of  the  gun.  The  rear  sight  was  usually  the  movable  one.  At  the  present  time  the 
most  approved  form  of  sight,  and  practically  the  only  one  in  use  in  the  navy,  is  that 


148 


EXTERIOR  BALLISTICS 


Theory   of 
bar  sights. 


in  which  a  telescope  has  been  substituted  for  the  old  pair  of  sights,  front  and  rear. 
This  telescope  is  so  mounted  that  it  is  capable  of  being  set  at  any  desired  angle  with 
the  axis  of  the  gun,  within  necessary  limits.  The  principles  involved  in  the  tele- 
scopic sight  are  the  same  as  in  the  old  bar  sights,  but  in  the  former  they  are  not  so 
clearly  apparent  or  so  easily  studied  as  in  the  latter.  For  this  reason  we  will  take  up 
the  theory  of  sights  from  the  point  of  view  of  the  old  system  of  bar  sights,  rear  sight 
adjustable,  and  the  application  of  these  theories  to  the  telescopic  sight  will  be  plainly 
apparent. 

220.  The  rear  sight  being  movable,  it  is  customary  to  graduate  its  bar  in  yards 
of  range  (and  sometimes  with  the  elevation  in  degrees  corresponding  to  the  range  in 
yards),  and  sometimes  there  is  added  the  time  of  flight  in  seconds  corresponding  to 
each  range,  this  last  information  being  for  use  in  setting  time  fuses  when  using 
shrapnel,  etc.  This  information  is  ordinarily  not  placed  on  the  range  scale  of  a 
telescopic  sight,  which  shows  only  the  range  in  yards;  and  if  such  information  be 
wanted  it  must  be  taken  from  the  range  table  for  the  gun  wJiich  is  now  furnished 
to  ships. 


AT' 


-.--H^' 


Figure  16. 


Sight  bar 
height. 


-J 


-yy„ 


221.  Figure  16  represents  the  usual  arrangement  of  bar  sights,  AC  being  the 
movable  graduated  rear  sight  bar,  at  right  angles  to  the  axis  of  the  gun,  and  B  the 
fixed  front  sight.  C'B  is  the  line  of  sight,  being  a  line  from  the  notch  in  the  rear 
sight  C"  to  the  top  of  the  front  sight  B,  and  CB  is  the  position  of  the  line  of  sight 
when  it  is  parallel  to  the  axis  of  the  gun,  the  rear  sight  notch  being  then  lowered  to 
C,  usually  its  lowest  position.  The  distance  GB^l  is  called  the  "  sight  radius  "  of 
the  gun,  and  the  line  GB  is  sometimes  called  the  "  natural  line  of  sight."  It  will  be 
seen  that  when  the  rear  sight  notch  is  raised  to  C",  and  the  line  of  sight  C'B  is  directed 
at  the  target  P,  the  axis  of  the  gun,  which  is  parallel  to  CB,  is  elevated  at  the  angle 
CBC  =  \\i,  or  the  angle  of  elevation  above  the  target.  As  we  will  deal  only  with  hori- 
zontal trajectories,  and  disregard  jump,  the  angle  of  departure  will  be  equal  to  the 
angle  of  elevation,  so  CBC'  =  ip  =  (j>.  The  distance  CC'  =  h  is  the  "  sight  bar  height " 
for  the  angle  of  departure  (f>,  and  it  is  evidently  given  by  the  equation 

h  =  l  tan  <t>  (194) 


VARIATION  OF  THE  TRAJECTORY  FROM  A  PLANE  CURVE     149 

When  the  heights  for  the  range  graduations  of  the  sight  bar  are  to  be  computed  the 
above  formula  is  used. 

222.  In  Figure  16  the  trajectory  is  represented  as  though  the  axis  of  the  gun 
coincided  with  the  line  of  sight,  instead  of  being,  as  it  really  is,  offset  from  it  by  at 
least  the  radius  of  the  breech  of  the  gun.  No  appreciable  error,  however,  results  from 
making  this  assumption  in  the  ordinary  use  of  guns,  except  in  the  case  of  turret 
guns,  where  the  pointers'  sights  may  be  located  at  a  considerable  distance  from  the 
axis  of  the  gun,  in  the  turret  or  in  the  pointers'  hoods.  In  this  case  this  distance 
must  be  allowed  for  in  figuring  on  the  fall  of  shot,  and  it  is  customary  in  bore  sight- 
ing these  guns  to  so  adjust  the  sights  that  the  line  of  sight  of  each  of  them  will 
intersect  the  axis  of  the  bore  prolonged  at  the  most  probable  fighting  range.  At  the 
proving  ground,  where  extreme  accuracy  is  necessary,  as  in  attacking  armor  plates, 
etc.,  it  is  customary  to  use  bore  sights  in  sighting  the  gun,  thus  eliminating  the  error 
due  to  the  offset  of  the  line  of  sight  of  the  regular  sights  from  the  axis  of  the  gun. 

223.  Besides  the  movement  of  the  rear  sight  up  and  down  to  enable  the  gun  to   sliding  leaf, 
be  pointed  with  the  proper  elevation,  it  is  desirable  to  have  some  means  of  moving 

it  sideways,  so  that  the  line  of  sight  may  be  adjusted  at  any  desired  angle  with  the 
axis  of  the  gun,  within  reasonable  limits,  in  the  horizontal  plane  as  well  as  in  the 
vertical.  This  is  for  the  purpose  of  allowing  for  drift  or  other  lateral  deviations  of 
the  projectile,  by  causing  the  gun  to  point  the  proper  amount  to  one  side  of  the  target 
at  which  the  line  of  sight  is  directed.  In  the  bar  sight  this  is  usually  done  by  forming 
the  rear  sight  notch  in  a  "  sliding  leaf,"  a  piece  mounted  on  the  head  of  the  sight  bar 
and  movable  by  suitable  mechanism  at  right  angles  to  the  sight  bar  and  to  the  axis 
of  the  gun. 

224.  In  Figure  16,  D  and  D'  represent  two  positions  of  the  sliding  leaf  on 
opposite  sides  of  the  central  position  C".  Evidently,  if  the  line  of  sight  B'B  be 
directed  at  the  target  P",  and  there  be  no  deviation  of  the  projectile  in  flight,  the 
latter  will  fall  at  P;  and  so  a  movement  C'V  =  d^  of  the  sliding  leaf  to  the  left  will 
cause  the  projectile  to  fall  P"P  =  D^  to  the  left  of  the  point  aimed  at  P".  And, 
similarly,  the  moving  of  the  sliding  leaf  C'D  =  d^  to  the  right  will  cause  the  projectile 
to  fall  P'P  =  D2  to  the  right  of  the  point  aimed  at,  P'.  Therefore  to  correct  for  a 
deviation  of  the  projectile  due  to  drift,  wind,  motion  of  the  gun  or  target,  or  any  other 
cause,  the  sliding  leaf  is  moved  in  the  opposite  direction  to  the  deviation,  and  we 
have  the  general  rule  : 

Move  the  sliding  leaf  (or  rear,  or  eye  end  of  the  telescope)  to  the  side  toivards 
which  you  ivish  the  projectile  to  go. 
Also,  for  the  relation  between  the  motion,  d,  of  the  sliding  leaf  and  the  resulting 
deviation,  D,  of  the  projectile,  we  have  from  similar  triangles 

^^_  FP  d  D 

C'B        PB    ^^  lseccl>   ~   X 

whence  d=  —^"^   V  (195) 

The  error  which  results  from  putting  see4>  =  l  in  (195)  is  inappreciable  for  the 
small  angles  of  departure  required  in  the  ordinary  use  of  modern  guns,  being  only 
one-half  of  1  per  cent  for  (f>  =  G° ;  and,  as  it  is  only  a  little  over  3  per  cent  for  (f>  =  lo°, 
the  limit  of  elevation  possible  with  our  usual  naval  gun  mounts ;  and,  as  the  value  of 
D  to  be  allowed  for  is  seldom  as  closely  known  as  that,  it  is  evident  that  in  direct  fire, 
under  all  ordinary  circumstances,  we  may  use 

d=~D  (196) 

*  D  and  X  must  be  in  the  same  units,  either  both  feet  or  both  yards,  d  will  then  come 
in  the  same  units  as  I,  usuallj^  inches. 


150 


EXTEEIOE  BALLISTICS 


Permanent 
angle. 


225.  It  was  common  practice  with  naval  guns  and  bar  sights  to  correct  for  the 
greater  part  of  the  drift  automatically  by  inclining  the  bar  sight  in  a  plane  per- 
pendicular to  the  axis  of  the  bore  of  the  gun,  so  as  to  make  with  the  vertical  an 
angle  i  called  the  "  permanent  angle."  Eef erring  to  Figure  17,  we  see  that  the  three 
points  D,  C  and  C  are  in  the  same  plane,  which  is  perpendicular  to  the  line  CB,  the 
angles  CC'D  and  BCD  being  right  angles ;  the  points  B,  C  and  C  are  all  three  in  the 
same  plane,  which  is  at  right  angles  to  DCC;  B,  C  and  D  are  in  the  same  plane, 
which  is  at  right  angles  to  BCC;  and,  similarly,  for  P,  P'  and  Q.    Then  we  have  that 


CB  =  l,  being  the  natural  line  of  sight,  and  CD  =  h  is  the  sight  bar  height  for  the 
angle  of  departure  ^,  and  is  now  given  by 

h  =  l tan <j>  sect  (197) 

Then  if  PF  be  the  drift  in  yards,  at  the  range  R,  from  the  similar  triangles  we 

HP'  T) 

have  :^  =  ^  ;  but  DC  =  h  smi  =  I  tan  <^  tan  i,  and  CB  =  I  sec  <^.     Therefore 
C  B        R 

^^^  "^  ^^"  ^    =sin  </.  tan  i=  -S-  ,  whence  we  have 


sec  ^ 


R 


tan  -1  = 


D 


R  sin  <^ 


(198) 


226.  If  D  were  proportional  to  X  sin  4>,  which  it  is  not  far  from  being,  setting 
the  sight  bar  at  the  permanent  angle  i  given  by  (198)  would  exactly  compensate  for 
drift  at  all  ranges.  Actually,  however,  D  increases  a  little  more  rapidly  in  propor- 
tion than  X  sin  ^,  and  so  the  sight  should  be  more  inclined  for  long  than  for  sliort 
ranges.    In  practice,  when  bar  sights  are  used,  it  is  customary  to  compute  the  value 


*  D  and  R  must  be  in  the  same  units.    In  the  above  equation  they  are  both  expressed 
in  yards. 


VAEIATION  OF  THE  TRAJECTORY  FROM  A  PLANE  CURVE     151 

of  i  for  an  assumed  average  fighting  range,  and  to  set  the  bar  at  that  permanent  angle, 
leaving  any  uncompensated  drift  at  other  ranges  to  be  corrected  by  setting  the  sliding 
leaf  while  using  it  to  correct  for  deviations  due  to  wind  and  speed. 

227.  The  telescopic  sight  is  now  almost  universally  adopted,  its  great  advantage  Telescopic 
being  that  with  it  the  line  of  sight,  which  is  the  optical  axis  of  the  telescope,  is  much 

more  clearly  defined  and  can  be  directed  with  much  greater  accuracy  than  is  the  case 
with  the  old  bar  sight.  Also,  as  the  eye  is  held  close  to  the  telescope  in  pointing, 
which  coaild  not  be  done  with  the  old  sights,  the  parallactic  errors,  which  it  was 
formerly  almost  impossible  to  avoid,  are  now  practically  eliminated.  The  theory 
of  the  telescopic  sight  in  no  way  differs  from  that  of  the  bar  sight  as  explained  in 
the  preceding  paragraphs,  however,  but  the  mechanical  features  of  the  bar  sight  are 
such  that  the  bar  itself  actually  establishes  the  system  of  triangles  with  which  we 
have  been  dealing  and  from  which  the  mathematical  relations  are  plainly  apparent. 
With  the  telescopic  sight,  however,  with  both  ranges  and  deflections  marked  on  rotary 
drums,  circular  discs,  etc.,  the  motion  of  which  is  transferred  to  the  telescope  itself  by 
gearing,  or  any  similar  devices  ordinarily  in  use,  the  triangles  are  not  readily  appar- 
ent, although  the  relations  arising  from  them  of  course  still  exist;  the  sight  radius 
in  the  bar  sight  being  replaced  in  the  telescopic  mounting  by  the  distance  from  the 
pivot  of  the  sight  yoke  to  the  circle  of  graduations  on  the  sword  arm,  that  is,  by  the 
radius  of  curvature  of  the  scale  on  the  sword  arm. 

228.  In  the  telescopic  sight  the  telescope  is  so  mounted  that  it  can  be  set  at  any 
desired  vertical  angle  with  the  axis  of  the  gun,  within  practical  limits,  thus  enabling 
the  gun  to  be  pointed  at  the  proper  elevation,  the  elevating  scale  being  marked  in 
yards  in  range  computed  for  the  corresponding  angle  between  the  line  of  sight  and 
the  axis  of  the  bore.  The  telescope  can  also  be  rotated,  within  reasonable  limits, 
about  its  vertical  axis,  which  corrects  for  deviation  in  the  horizontal  plane  exactly 
as  did  setting  over  the  sliding  leaf  of  the  bar  sight ;  the  rotation  of  the  telescope  about 
its  vertical  axis  being  recorded  on  the  deflection  scale,  which  is  marked  in  "  knots," 
the  knots  thus  indicated  corresponding  to  speed  of  target.  The  reasons  for  this 
graduation  and  the  method  of  using  it  will  be  explained  later,  in  the  chapter  describ- 
ing the  use  of  the  range  tables.  Drift  is  not  compensated  for  in  the  mounting  of  this 
sight,  but  as  the  gun  is  elevated  the  pointer  on  the  deflection  scale  moves  up  or  down 
over  the  scale,  and  the  line  on  the  scale  for  each  knot  setting  of  the  sight  in  deflec- 
tion, instead  of  being  a  straight  line,  is  a  curve  so  computed  and  laid  on  the  scale 
that  when  the  deflection  pointer  is  on  it  the  drift  is  compensated,  no  matter  what 
the  range  may  be.  This  system,  while  not  automatic,  gives  perfect  compensation  for 
drift  at  all  ranges,  which  the  old  permanent  angle  system  did  not,  as  we  have  already 
seen,  and  introduces  no  troubles  or  errors  into  the  actual  process  of  setting  the  sight 
which  would  not  exist  without  it.  (See  Appendix  C  for  a  description  of  the  system 
of  arbitrary  deflection  scales  now  in  use  for  turret  guns.) 


153 


EXTEEIOR  BALLISTICS 


EXAMPLES. 
1.  Compute  the  drift  in  yards  for  the  following  conditions,  taking  the  angle  of 
departure  and  the  maximum  ordinate  from  the  range  tables.    Conditions  standard. 


DATA. 

ANSWERS. 

Problem. 

Projectile. 

Velocity. 

f.  s. 

Range. 

Yds. 

Drift. 

d. 
In. 

to. 
Lbs. 

c. 

Yds. 

A 

3 
3 
4 
5 
5 
6 
6 
6 
7 
7 
8 

10 
10 
12 
13 
13 
14 
14 

13 

13 

33 

50 

50 

105 

105 

105 

165 

165 

260 

510 

510 

870 

1130 

1130 

1400 

1400 

1.00 
1.00 
0.67 
1.00 
0.61 
0.61 
1.00 
0.61 
1.00 
0.61 
0.61 
1.00 
0.61 
0.61 
1.00 
0.74 
0.70 
0.70 

1150 
2700 
2900 
3150 
3150 
2600 
2800 
2800 
2700 
2700 
2750 
2700 
2700 
2900 
2000 
2000 
2000 
2600 

2500 

4000 

3500 

4000 

3700 

10500 

3800 

3200 

7000 

6700 

8000 

10000 

11000 

22000 

10000 

11000 

14000 

14500 

7.5 

B 

8.5 

c 

2.4 

D 

5.8 

E 

3.3 

F 

69.4 

G 

3.2 

H 

2.6 

I 

17.6 

J .... 

K      

15.7 
21.5 

L     

33.9 

M        

44.3 

N 

0 

233.8 
52.1 

P 

82.9 

0 

148.1 

R 

89.3 

2.  Compute  the  sight  bar  heights  in  inches,  and  the  distance  in  inches  that  the 
sliding  leaf  must  be  set  over  for  the  data  given  in  the  following  table,  conditions 
being  standard. 


DATA. 

ANSWERS. 

Problem. 

Gun. 

Range. 
Yds. 

<(>■ 

Sight  ra- 
dius or 
radius  of 
curva- 
ture 
of   sword 
arm. 
In. 

Deflection 
to  be  com- 
pensated. 

Yds. 

Sight 

bar 
height. 

In. 

Set  of  sliding 
leaf. 

Cal. 
In. 

I.T. 

f.  s. 

o 

Right 

or 

left. 

A 

B 

C 

D 

E 

F 

G 

H 

I   

J 

K 

L 

M 

N  ......  . 

0 

P 

Q 

R 

3 
3 
4 
5 
5 
6 
6 
6 
7 
7 
8 

10 
10 
12 
13 
13 
14 
14 

1150 
2700 
2900 
3150 
3150 
2600 
2800 
2800 
2700 
2700 
2750 
2700 
2700 
2900 
2000 
2000 
2000 
2600 

1700 

3600 

3100 

4000 

4500 

13500 

4100 

3400 

7000 

6700 

7800 

9700 

10900 

23500 

9600 

10800 

13600 

14000 

4°  19'  06" 
2     48    54 
1     18    42 
1     53    12 

1  45    48 

12  20    18 

2  05    18 
1     25    42 

4  44    48 

3  28    48 
3     51    12 
6     11    42 

5  48    18 
14     12    12 
10     03    18 
10     32    42 

13  54    00 
8     14    06 

24.750 
28.625 
45.900 
58.500 
42.625 
42.625 
42 . 625 
42.625 
55.850 
62.500 
41 . 125 
44.675 
44.675 
47.469 
61.094 
61.094 
36.219 
36.219 

25  R. 

20  L. 

40  L. 

50  R. 

30  R. 

75  L. 

50  R. 

30  L. 
100  R. 

75  L. 

50  R. 

60  L. 

50  R. 
150  L. 

70  R. 

60  R. 

50  L. 

75  L. 

1.869 
1.408 
1.051 
1.927 
1.312 
9.324 
1.5.54 
1.063 
4.637 
3.801 
2.770 
4.849 
4.542 
12.014 
10.833 
11.373 
8.963 
5.242 

0.365 
0.159 
0.592 
0.732 
0.284 
0.242 
0.520 
0.376 
0.801 
0.701 
0.264 
0.278 
0.206 
0.313 
0.452 
0.345 
0.137 
0.196 

Left 

Right 

Right 

Left 

Left 

Right 

Left 

Right 

Left 

Right 

Left 

Right 

Left 

Right 

Left 

Left 

Right 

Right 

Note. — The  data  given  in  the  sight  radius  column  is  approximate  only,  and  must  not 
be  accepted  as  reliable.  The  computed  sight  bar  heights  and  the  set  of  the  sliding  leaf  are 
for  the  old  bar  sight.  To  use  them  for  the  telescopic  sight  they  must  be  transformed  as 
necessary  into  the  proper  distances  for  marking  on  sword  arm  scales  or  range  or  deflec- 
tion scales,  as  the  case  may  be.  These  supplementary  computations  have  to  do  with  the 
mechanical  features  of  the  sight  only,  and  not  with  the  principles  of  exterior  ballistics, 
and  are  therefore  not  considered  here. 


CHAPTER  14. 

THE  EFFECT  OF  WIND  UPON  THE  MOTION  OF  THE  PROJECTILE.  THE 
EFFECT  OF  MOTION  OF  THE  GUN  UPON  THE  MOTION  OF  THE  PRO- 
JECTILE. THE  EFFECT  OF  MOTION  OF  THE  TARGET  UPON  THE  MOTION 
OF  THE  PROJECTILE  RELATIVE  TO  THE  TARGET.  THE  EFFECT  UPON 
THE  MOTION  OF  THE  PROJECTILE  RELATIVE  TO  THE  TARGET  OF  ALL 
THREE  MOTIONS  COMBINED.  THE  COMPUTATION  OF  THE  DATA  CON- 
TAINED IN  COLUMNS  13,  14,  15,  16,  17  AND  18  OF  THE  RANGE  TABLES. 

New  Symbols  Introduced. 

W .  . .  .  Eeal  wind,  force  in  feet  per  second, 
/?....  Angle  between  wind  and  line  of  fire. 
Wx .  . .  •  Component  of  W  in  line  of  fire  in  feet  per  second. 
Wi2a!-  •  •  •  Wind  component  of  12  knots  in  line  of  fire  in  feet  per  second. 

Wz.  . .  .  Component  of  W  perpendicular  to  line  of  fire  in  feet  per  second. 
TFi2«-  •  •  •  Wind  component  of  12  knots  perpendicular  to  line  of  fire  in  feet  per 
second. 
X .  . .  .  Range  in  feet  without  considering  wind. 
X' .  . . .  Range  in  feet  considering  wind. 

V .  . .  .  Initial  velocity  in  foot-seconds  without  considering  wind. 
y .  . .  .  Initial  velocity  in  foot-seconds  considering  wind. 
<^ .  . .  .  Angle  of  departure  without  considering  wind. 
<f>' .  .  .  .  Angle  of  departure  considering  wind. 
T .  .  .  .  Time  of  flight  in  seconds  without  considering  wind. 
T" .  . .  .  Time  of  flight  in  seconds  considering  wind. 
AAV-  •  •  •  Variation  in  range  in  feet  due  to  Wi. 
AA'^ott-  •  •  •  Variation  in  range  in  feet  due  to  a  wind  component  of  12  knots  in 
line  of  fire. 
ABw  •  •  •  Variation  in  range  in  yards  due  to  Wx- 
LR^ow-  •  •  •  Variation  in  range  in  yards  due  to  a  wind  component  of  12  knots  in 
line  of  fire. 
y . . .  .  Angle  between  trajectories  relative  to  air  and  relative  to  ground. 
Dw-  •  •  •  Deflection  in  yards  due  to  wind  component  Wg  perpendicular  to  line 
of  fire. 
D^^w-  •  •  •  Deflection  in  yards  due  to  wind  component  of  12  knots  perpendicular 
to  line  of  fire. 
G.  . .  .  Motion  of  gun  in  feet  per  second. 
Gx-  •  •  ■  Component  of  G  in  line  of  fire  in  feet  per  second. 
G'lox-  .  •  •  Motion  of  gun  of  12  knots  in  line  of  fire  in  feet  per  second. 

Gz.  .  .  .  Component  of  G  perpendicular  to  line  of  fire  in  feet  per  second. 
Gjos.  . .  .  Motion  of  gun  of  12  knots  perpendicular  to  line  of  fire  in  feet  per 

second. 
AXg  ....  Variation  in  range  in  feet  due  to  Gx. 
AX120.  •  •  •  A'ariation  in  range  in  feet  due  to  a  motion  of  gun  in  line  of  fire  of 
12  knots. 
Ai^G.  .  .  .  Variation  in  range  in  yards  due  to  Gx- 
AR^2G-  -  • .  Variation  in  range  in  yards  due  to  a  motion  of  gun  in  line  of  fire 
of  12  knots. 


154  EXTEEIOE  BALLISTICS 

Dg  •  •  •  .  Deflection  in  yards  due  to  a  motion  of  gun  of  Gz  perpendicular  to 
line  of  fire. 
Di20-  •  •  •  Deflection  in  yards  due  to  a  motion  of  gun  of  12  knots  perpendicular 
to  line  of  fire. 
T .  . . .  Motion  of  target  in  feet  jser  second. 
Tx-  .  ■ '  Motion  of  target  in  line  of  fire  in  feet  per  second. 
Tiaa-  •  •  •  •  Motion  of  target  of  12  knots  in  line  of  fire  in  feet  per  second. 

Tg.  . . .  Motion  of  target  perpendicular  to  line  of  fire  in  feet  per  second. 
Ti2« ....  Motion  of  target  of  12  knots  perpendicular  to  line  of  fire  in  feet  per 

second. 
AXt  ....  Variation  in  range  in  feet  due  to  T. 
AX^2T'  • .  •  Variation  in  range  in  feet  due  to  a  motion  of  target  of  12  knots  in 
line  of  fire. 
ARt-  . .  .  Variation  in  range  in  yards  due  to  T. 
AR^2t-  '  •  •  Variation  in  range  in  yards  due  to  a  motion  of  target  of  12  knots  in 
line  of  fire. 
Dt  ....  Deflection  in  yards  due  to  a  motion  of  target  Ts  perpendicular  to 
line  of  fire. 
DisT.  • .  .  Deflection  in  yards  due  to  a  motion  of  target  of  12  knots  perpen- 
dicular to  line  of  fire. 
a. . . .  Angle  of  real  wind  with  course  of  ship. 
a' .  . . .  Angle  of  apparent  wind  with  course  of  ship. 
TFj ....  Velocity  of  real  wind  in  knots  per  hour. 
TFo  •  •  •  •  Velocity  of  apparent  wind  in  knots  per  hour. 

Section  1. — The  Effect  of  ^Yind  Upon  the  Motion  of  the  Projectile. 

229.  In  considering  the  effect  of  wind  upon  the  flight  of  the  projectile,  we  are 
obliged,  for  want  of  a  better  knowledge,  to  assume  that  the  air  moves  horizontally 
only,  and  that  its  direction  and  velocity  are  the  same  throughout  the  trajectory  as 
we  observe  them  to  be  at  the  gun.  Actually  the  wind  is  never  steady,  either  in  force 
or  in  direction ;  its  velocity  usually  increases  with  the  height  above  the  gun,  and  its 
motion  is  not  always  confined  to  the  horizontal  plane.  Moreover,  lateral  wind 
pressure  alters  the  drift  due  to  rotation. 

230.  It  is  for  these  reasons  that  the  deviations  caused  by  the  wind  can  only  be 
roughly  approximated;  and,  consequently,  that  experiments  for  determining  any  of 
the  ballistic  constants,  to  be  of  value,  must  be  made  when  it  is  calm  or  very  nearly  so. 

Primary  231,  Let  US  denote  by  W  the  velocity  of  the  wind  in  feet  per  second,  and  by 

YVx  and  YV~,  respectively,  the  components  of  that  velocity  in  and  at  right  angles  to  the 
plane  of  fire.  Also  let  us  call  Wx  positive  when  it  is  with  the  fiight  of  the  projectile, 
and  negative  when  it  is  against  it.  Let  us  also  call  Wz  positive  when  it  tends  to  carry 
the  projectile  from  right  to  left  of  an  observer  looking  from  gun  to  target,  and 
negative  in  the  opposite  case.  In  Figure  18  let  us  denote  by  ji  the  angle  between  the 
direction  from  gun  to  target  and  the  direction  towards  which  the  wind  is  blowing, 
measuring  the  angle  to  the  left  from  the  first  direction  around  to  the  second. 

Then  in  Figure  18(a),  fi  is  in  the  first  quadrant,  and  \Yx  is  blowing  with  the 
projectile  and  is  positive,  and  Wz  causes  lateral  motion  to  the  left  and  is  also  positive. 
In  Figure  18(&),  ^  is  in  the  second  quadrant,  and  Wx  is  negative  and  Wz  is  positive. 
In  Figuro  18(c),  )8  is  in  the  third  quadrant,  and  both  Wx  and  Wz  are  negative.  In 
Figure  18 ((i),  p  is  in  the  fourth  quadrant,  and  Wx  is  positive  and  Wz  is  negative. 
Note  especially  that,  in  the  system  of  notation  adopted,  fi  is  the  angle  between  the 
plane  of  fire  and  the  direction  towards  and  not  that  from  which  the  wind  is  blowing. 


VARIATION  OF  THE  TRAJECTORY  FROM  A  PLANE  CURVE     155 

(In  a  later  chapter  dealing  with  practical  service  problems  concerning  the  wind,  it 
will  be  found  that  the  wind  is  generally  stated  as  coviing  from  a  given  direction,  but 
the  reverse  convention  is  used  in  this  chapter.  Bear  the  difference  constantly  in  mind 
and  do  not  be  confused  by  it.) 


J^^ 


(a) 

Wx  is  + 
Wz  is  + 


(6) 

Wx  is  — 
Wz  is  + 


.'7  ^. 


,--/S^ 


(c) 

id) 

Wx  is  — 

Wx  is  # 

Wz  is  — 

We  is  — 

Figure  18. 

>~^ 


232.  By  an  examination  of  the  figure  we  see  that,  no  matter  what  the  direction 
of  the  wind,  we  shall  always  have  its  components  in  the  two  primary  reference  planes 
given  with  their  proper  signs  by  the  two  expressions: 

F^  =  Fcos^  (199) 

W„,  =  Fsin^  (200) 

233.  The  respective  values  of  the  wind  components  being  thus  found,  they  are 
hereafter  treated  as  though  independent  of  each  other,  each  producing  its  own  effect 
in  its  own  primary  plane  only. 

234.  To  find  the  effect  of  the  wind  upon  the  range,  let  ^Vx  be  the  wind  com- 
ponent in  the  plane  of  fire  in  feet  per  second,  positive  when  with  the  flight  of  the  pro- 
jectile and  negative  when  against  it ;  let  V  and  <^  be  the  initial  velocity  and  angle  of 
departure,  respectively,  relative  to  still  air  or  to  the  ground,  which  is  of  course  station- 
ary; and  let  X  be  the  range  in  feet  for  this  initial  velocity  and  angle  of  departure, 
that  is,  relative  to  the  ground  or  to  still  air. 

235.  Let  us  now  designate  by  X',  V  and  ^'  the  range,  initial  velocity  and  angle 
of  departure,  respectively,  relative  to  the  moving  air,  of  a  projectile  fired  with  a  range 
of  X,  initial  velocity  of  V,  and  angle  of  departure  of  4>  relative  to  still  air  or  to  the 
ground,  and  by  dV  and  dct>  the  differences  between  V  and  V  and  <^  and  <}>',  respect- 
ively. Let  us  designate  by  T"  and  T  the  corresponding  times  of  flight.  Now  let  us 
suppose  that  the  wind  component  along  the  line  of  flight,  Wx,  is  blowing  with  the 
flight  of  the  projectile;  in  which  case,  while  the  projectile  is  in  flight,  the  air  through 
which  it  is  traveling  moves  in  the  same  direction,  carrying  the  projectile  with  it  a 
distance  WxT'.  Then  the  total  horizontal  distance  traveled  by  the  projectile  relative 
to  the  ground  or  to  still  air  will  be  X'  +  WxT'.  As  the  normal  range  relative  to  the 
ground  corresponding  to  V  and  ^  is  X,  then  the  difference  between  the  two  ranges,  or 
the  change  in  range  caused  by  the  wind,  would  be 

^X^X'  +  WxT'-X  (201) 


Effect  in 
range. 


156 


EXTEEIOE  BALLISTICS 


Force 
diagrams. 


236.  In  order  to  use  the  above  equation  it  is  necessary  to  determine  the  values  of 
X'  and  T',  and  we  can  do  this  by  methods  previously  explained  if  we  can  determine 
the  corresponding  values  of  V  and  (f)'.  This  we  can  do  if  we  can  find  the  values  of 
dV  and  dcf).  To  do  this  let  us  draw  the  triangle  of  forces  acting  in  this  case  (and  also 
for  a  negative  wind) .    We  would  have  the  results  as  shown  in  Figure  19. 

Figure  19  (a)  is  for  a  positive  wind,  for  which  Wx  is  positive  (being  drawn  in  the 
proper  direction  for  constructing  a  triangle  of  forces  with  all  parts  of  proper  relation 
to  one  another),  and  from  the  diagram  it  will  be  seen  that  OA  =  V  combined  with 


H^^c/ 


AB—  TF^  gives  05=  F,  which  is  less  than  OA  =  "T  by  the  amount  dY  =  AC—\\'x  cos  <^. 
Also  the  angle,  BOH  =  (^'  is  greater  than  the  angle  AOH  =  cf>  by  the  angle 

d^  =  AOB=^  =  ^^^^ 

(assuming  that  for  this  small  angle  the  sine  and  the  circular  measure  of  the  angle  are 
equal).  In  other  words,  the  forces  acting  would  produce  a  trajectory  relative  to  the 
moving  air  for  which  the  initial  velocity  is  V'  =  V  —  dV  and  the  angle  of  departure  is 
<f>'  =  (f>  +  d(l>.  Similarly,  from  Figure  19(&),  where  Wx  is  negative  or  against  the  flight 
of  the  projectile,  we  would  have  V  greater  than  V  by  the  amount  dV=Wx  cos  (j), 

and  cf>'  less  than  4>  by  the  amount  f/0  =  -^  =  ^''^^  ^  .    Thus,  in  both  cases  we  can 

obtain  the  values  of  the  changes  in  V  and  0  with  their  proper  signs  from  the 
expressions 

dV=-Wx  cos  cf>  (202) 

d<f>=^^^^  (203) 

The  negative  sign  is  arbitrarily  introduced  into  the  second  term  of  (202)  to  ensure 
that  a  positive  value  of  Wx  shall  always  produce  a  negative  value  of  dV,  and  that  a 
negative  value  of  Wx  shall  always  produce  a  positive  value  of  dV,  as  is  seen  from 
the  triangles  of  forces  must  always  be  the  case. 

237.  To  determine  the  effect  of  a  wind  Wx,  therefore,  we  compute  dV  by  (202) 
and  d(f>  by  (203) ;  compute  the  range  Z  given  by  V  and  ^  by  methods  heretofore 
explained;  compute  the  range  X'  and  the  time  of  flight  T'  given  by  V'=V  +  dV  and 
<^'  =  (ji  +  d(i>  by  methods  heretofore  explained,  and  then  by  (201)  we  can  compute  the 
change  in  range  due  to  the  wind. 


VARIATION  OF  THE  TRAJECTORY  FROM  A  PLANE  CURVE     157 

238.  An  examination  of  Figure  20  will  help  to  reach  a  clear  understanding  of  the 
foregoing  method  of  determining  the  effect  of  wind  upon  the  range.  Let  0  represent 
the  stationary  gun  and  M  the  stationary  target  which  would  be  hit  if  there  were  no 
wind  at  the  range  OM  =  X,  by  a  projectile  fired  from  0  with  an  initial  velocity  V  and 
an  angle  of  departure  4>,  the  flight  being  through  still  air  or  considered  relative  to 
the  ground  with  no  wind  Mowing.  Then  OSH  represents  the  trajectory  relative  to 
still  air  or  to  the  ground;  that  is,  the  trajectory  that  would  be  given  in  still  air  by  the 
initial  velocity  V  and  the  angle  of  departure  <^',  in  which  the  projectile  would  fall 
short  of  M  by  the  distance  HM.  But  since  the  air  is  moving  with  the  projectile  with 
a  velocity  Wx,  and  is  carrying  the  projectile  with  it  with  the  same  velocity  during  the 
time  of  flight  T,  when  the  projectile  reaches  the  ground  the  trajectory  OSH  will 
have  moved  to  the  position  O'S'W,  the  actual  point  of  fall  will  be  at  //',  and  the 


>^A^  >: 


Figure  20. 


actual  range  over  the  ground  will  be  OH'  =  X'  -\-^VxT'.    Therefore,  instead  of  falling 
at  the  target  M,  the  projectile  will  really  strike  at  the  point  H',  a  distance  beyond  M  of 

^X  =  X'+^Ya:T'-X 

239.  The  process  just  explained  is  not  only  a  somewhat  lengthy  and  inconvenient 
one,  but  the  methods  of  interpolation  used  with  the  ballistic  tables  were  not  devised 
with  this  particular  process  in  view,  and  do  not  produce  results  sufficiently  accurate 
to  determine  the  small  differences  in  range  with  the  precision  necessary  in  this  class 
of  problem.*  A  carrying  out  of  the  process  just  described  may  therefore  not  bring 
correct  results,  and  it  is  desirable  to  reduce  the  formulae,  if  possible,  to  some  form 
more  convenient  for  practical  use  and  that  will  not  involve  the  use  of  the  ballistic 
tables.  Although,  as  already  stated,  the  above  formulge  are  not  useful  for  obtaining 
practical  results  in  this  case,  they  are  nevertheless  theoretically  correct,  and  from 
them  will  now  be  derived  the  formula  that  is  actually  used  in  practice  in  computing 
the  data  contained  in  Column  13  of  the  range  tables.  From  Chapter  4  we  see  that 
(when  a  =  2,  which  is  sufficiently  accurate  for  present  purposes)  the  relation  between 
the  range  in  air  and  the  angle  of  departure  is 

-  9J^  -  9^ 
y2         V' 


sin2</>: 


(I  +  PZ) 


(204) 


Taking  logarithmic  differentials  of  this  expression,  that  is,  differentiating  and  then 
dividing  by  the  original  equation,  and  considering  <f>  and  X  as  the  only  variables  (the 
small  angle  2(j>  being  considered  as  having  its  natural  sine  equal  to  its  circular 
measure)  we  get 

OJj.  1     I    41.  V  /7V 

(205) 


2dcf>    _l+^kX  ^  dX 


tan2</>       l+^JcX       X 


*  See  foot-note  to  paragraph  153,  Chapter  8.  An  effort  to  use  the  formulae  just  derived 
for  the  value  of  the  effects  of  wind  upon  the  flight  of  the  projectile,  by  the  use  of  Ingalls' 
method,  with  the  interpolation  formulae  given  in  Chapter  8,  will  not  be  successful,  because 
those  interpolation  formulae  neglect  second  and  higher  differences;  and  the  limits  of 
accuracy  within  which  these  results  would  have  to  be  obtained  in  this  case  are  too  narrow 
to  permit  such  higher  differences  than  the  first  to  be  neglected  in  using  Table  II. 


158  EXTERIOR  BALLISTICS 

Now  l  +  ^kX  =  n;  whence  l  +  |-^•.Y  =  2?^— 1,  and  so  (205)  becomes 

2dci>     _  2n-l  ^  dX 
tan  24>  n  X 

whence  —  =     ^^      X  -^^  (206) 

^  ^®  Z        2n-l       tan2(;S  ^        ^ 

240.  In  this  and  in  similar  expressions  the  value  of  d(j)  must  of  course  be  ex- 
pressed in  circular  measure  (l'  =  . 0002909),  but  when  the  form  is  the  ratio  -^  ,  the 

value  is  the  same  whether  (/<^  and  4>  are  expressed  in  minutes  of  arc  or  in  circular 
measure.    Now  returning  to  (204)  and  writing  it  in  the  form 

V^sin2cb  =  gX{l  +  ikX) 
and  taking  logarithmic  differentials  with  regard  to  V  and  X  as  variables,  we  get 

2dV  _l  +  ^kX  ^dX  .on7\ 

~T^-T+pZ^   X  ^^"^^^ 

which,  substituting  n  for  l  +  p-.Y  and  2n  —  l  for  1  +  ^hX,  becomes 

24L  We  have  found  in  (208)  an  expression  for  the  variation  in  range  due  to  a 

variation  in  initial  velocity,  and  in  (206)  an  expression  for  the  variation  in  range 

due  to  a  variation  in  the  angle  of  departure ;  and  we  have  already  seen  that  the  effect 

upon  the  range  of  a  wind  component  in  the  plane  of  fire  is  to  give,  so  far  as  results 

are  concerned,  an  apparent  variation  in  both  V  and  4>-    Therefore  Z'  —  Z  is  nothing 

but  the  change  in  range  which  would  result  from  increasing  V  hj  dV  and  <^  by  c?(^, 

dV  being  determined  by  (202)  and  d<f>  by  (203).    Then  by  employing  (208)  and 

(206)  we  see  that  the  change  in  Z  due  to  simultaneous  changes  dV=  —Wx  cos  ^  and 

Wx  sin  <f>  ■      ■        1     ±1 
dcj)  =  — ^, — —  IS  given  by  the  expression 

Wx  sin  cf)  _  Wx  cos  <^\  ('909') 

2n-l\V'taji2^  V       )  ^''     ' 

Now  we  may  put  V  for  V  in  the  preceding  expression  without  material  error,  because 
jy  is  always  very  small  compared  with  V^  and  the  expression  then  reduces  to 

X'-X  2n     ^WJsm<f>       _    . 

AT  sin  (b  ,  ,  (  tan  d>       -, 

Now  T ^  -  cos  </)  =  cos  <^   T ^  - 1 

tan  2(^  \tan2<^ 

T  ^      o  J      sin  2<i        2  sin  (^  cos  (h 

and  tan  2<i>  = ^  =  — ,  .       •   ., 

cos  2<^       cos-<^  — sm-^ 

1                               ,  /  tan  d)       -,\  ,  /sin  d>  ^  cos- <^  — sin- <^      ., 

whence  cos  (/>  ( -^ ~j~  —  1  I  =  cos  <^  (  — ^  X    ^    .     .    — r  —  1 


tan  2<^        /  \cos<^         2  sin  <^  cos  ^ 

,  /cos-  <i  — sin-  <b      -,\ 
^^^^-^i       2cos^<^        ^^1 

=  £^(l-tan^.^-2) 

whence  ;^^  -cos<A  =  ^  (-tan^  ^-1)  (310) 

tan  2<j)  2  -r       / 


VAEIATION  OF  THE  TRAJECTORY  FROM  A  PLANE  CURVE     159 

Therefore,  from  the  above,  we  get 

X'-X n_        Wr  cos  <i>  ,     ,      ,         ^s 

whence,  neglecting  tan-  c^  in  comparison  with  unity, 

'2n  —  l  V 

and  substituting  this  in  (201)  we  finally  get  for  the  change  in  range  due  to  the  wind 
component  Wx 

AX=W,  (t--^^  X  ^^^)  *  (311) 

In  the  above  equation,  T,  although  actually  the  time  of  flight  for  V  and  4>',  may  be 
taken  as  the  time  of  flight  for  the  actual  firing  data,  V  and  <l>,  without  introducing 
any  material  errors.  This  formula  is  the  one  employed  in  computing  the  data  in 
Column  13  of  the  range  tables,  giving  the  change  in  range  resulting  from  a  wind  com- 
ponent of  12  knots  in  the  plane  of  fire. 

242.  ^STow  let  us  compute  the  data  for  that  column  for  our  standard  problem, 
the  12"  gun,  7  =  2900  f.  s.,  w  =  870  pounds,  c=0.61,  i2  =  10,000,  r  =  12.43,  and 
</>  =  4°  13'  14".     We  have  the  formula  given  in  the  preceding  paragraph  and 

r-sin2<i        T    „/  12x6080  -,  . 

n=  -^^  and   W,,.=  60x60x3  ^^'^'  ?'"  '^'""^ 

7  =  2900    log  3.46239 2  log  6.92480 

2<^  =  8°  26'  28"    sin  9.16669-10 

^  =  32.2    log  1.50786 colog  8.49214-10 

Z  =  30000    log  4.47712 colog  5.52288-10 

/i=  1.278    log  0.10651 

2n  =  2.556 

2n- 1  =  1.556    log  0.19200 colog  9.80800-10 

n  =  1.278    log  0.10651 

Z  =  30000    log  4.47712 

(^  =  4°   13'  14"    cos  9.99882-10 

7  =  2900    log  3.46239 colog  6.53761-10 

8.47    log  0.92806 

r=  12.43 


3.96 log  0.59770 

12    log  1.07918 

6080    log  3.78390 

60x60x3  =  10800   log  4.03342 colog  5.96658-10 

Ai?,2TF  =  26.7  yards    log  1.42736 

*  The  above  is  the  formula  actually  employed.  There  seems  to  be  no  good  reason, 
however,  for  neglecting  tan'  0,  for  tan'  0  +  1  =  sec'  </>,  and  if  we  substitute  this  value, 
instead  of  dropping  the  tan'  (p,  we  would  get  as  a  final  result 


^  =  ^^^(^-27^x^^r^) 


which  is  equally  easy  for  work,  and  is  more  in  keeping  with  the  form  of  the  expression  for 
determining  the  deflection  due  to  wind  given  in  equation  (212).  The  difference  in  results 
is  not  material  however. 


IGO 


EXTERIOR  BALLISTICS 


Lateral  243.  To  determine  the  lateral  deviation  due  to  wind,  let  Tf  ^  be  the  lateral  wind 

aeviation 

due  to  wind,  component  in  foot-seconds,  positive  when  it  blows  from  right  to  left  across  the  line 
of  fire,  and  negative  when  it  blows  in  the  reverse  direction ;  let  V  and  <^  be  the  initial 
velocity  and  angle  of  departure  relative  to  still  air  or  to  the  ground,  and  Z  be  the 
corresponding  range,  that  is,  the  range  wheri  there  is  no  wind.  Then  if  we  draw  the 
triangle  of  forces  for  this  case,  we  may  obtain  the  initial  velocity  and  direction  of 
flight  relative  to  the  moving  air.  Thus  referring  to  Figure  21(a),  which  represents 
the  case  of  a  negative  wind,  the  resultant  of  OA  =  V  with  OC=  —  Ws  is  OB=V', 
which  is  very  slightly  greater  than  V;  the  angle  BOD  =  (j>',  which  V  makes  with  the 
horizontal,  is  very  slightly  less  than  AOE  =  (f>;  and  V  is  inclined  to  the  left  so  as  to 
make  with  V  the  small  angle  BOA, 

z  ^ 


yi-tan  y 


(b) 
Figure  31. 


244.  Now  since  Y'  and  </>'  differ  so  little  from  Y  and  <^,  and  since  the  effect  of 
the  increase  in  Y  is  offset  by  that  of  the  decrease  in  ^,  we  may  take  the  range  X' 
corresponding  to  Y' ,  <f)',  to  be  practically  the  same  as  the  range  X  corresponding  to 
Y,  </>.  Therefore  the  only  essential  difference  between  the  trajectory  relative  to  the 
moving  air  and  that  relative  to  the  ground  or  still  air  is  that  the  plane  of  the  former 
makes  the  angle  DOE  =  y  with  the  plane  of  the  latter.  Referring  now  to  Figure 
21(h),  in  which  0  represents  the  gun  and  M^  the  target  at  the  range  OMq  =  X,  we 


see  that  tan  y: 


;  and  OM'  is  the  horizontal  trace  of  the  trajectory  relative 


Y  cos  <f> 

to  the  moving  air.  But  while  the  projectile  moves  through  the  air  from  0  to  M',  the 
air  itself  has  moved  WzT  to  the  right,  and  so  the  projectile  really  strikes  to  the  right 
of  the  target  by  the  distance 


Mjr  =W,T-X  tan  y=  W,  It-  ^ — \ 

\         Y  cos  <^/ 


Thus  the  lateral  deviation  caused  by  the  wind  component  VF«  normal  to  the  line  of 
fire  is  given  by  the  expression 

X 


Dw=W.    T 


Y  cos  <^ 


(212) 


VARIATION  OF  THE  TRAJECTORY  FRO^^I  A  I'LANE  CURVE     161 

in  which  T,  though  really  the  time  of  flight  corresponding  to  V,  </>',  mav  without 
appreciable  error  be  taken  as  the  time  of  flight  for  the  actual  firing  data. 

245.  Now  let  us  return  to  our  standard  problem  and  find  the  data  for  Column 
14  in  the  range  table;  deviation  for  lateral  wind  component  of  13  knots;  for  our  12" 

gun  at  10,000  yards.   We  have  the  above  formula,  and  also  W^.^=  ^'^^^^  "^l  vards 
per  second. 


GO  X  60x3 


X  =  30000    log  4.47712 

7  =  2900    log  3.4G240 colog  6.53760-10 

<^  =  4°   13'  14"    sec  0.00118 

10.37    log  1.01590 

T=  12.43 

2.06    log  0.31387 

12    log  1.07918 

6080    log  3.78390 

60x60x3  =  10800    log  4.03342 coW  5.96658-10 


Djow  =  13.9  yards log  1.14353 

Section  2. — The  Effect  of  Motion  of  tlie  Gun  Upon  the  Motion  of  the  Projectile. 

246.  As  in  the  case  of  the  wind,  we  resolve  the  horizontal  velocity  of  the  gun 
due  to  the  ship's  motion  into  two  components,  Gx  in  the  plane  of  fire,  and  Gg  at  right 
angles  to  that  plane;  and  determine  their  effects  separately,  the  first  affecting  the 
range  only  and  the  second  the  lateral  deflection  only. 


P---- 


Figure  22. 


247.  Let  Gx  be  the  resolved  part  of  the  speed  in  the  line  of  fire  in  foot-seconds, 
positive  when  with  and  negative  when  contrary  to  the  flight  of  the  projectile.  Then 
evidently  the  true  initial  velocity  of  the  projectile  is  the  resultant  of  Gx  and  V,  and, 
as  shown  in  Figure  22,  7  receives  the  increment  i^V=Gx  cos  </>,  while  4>  is  decreased 

by  A<^  =  ^^^ — ^.     But  by  equations  (208)  and  (206),  these  two  changes  in  7  and 
<^,  respectively,  will  cause  a  change  in  range  given  by 


Change  in 
range  due 
to  motion 
of  gun, 


AXg_      2n 
X         2n-l 


Gx  cos  4>  _   Crx  sin  <^\ 
7  7  tan  2cf>l 


AXo 
X 


2n  Gx  cos  ^  /g  _  2  tan  <^\ 

2/i-l  7       \        tan  2c}> ) 


11 


162  EXTEEIOE  BALLISTICS 

-KT  J.      o  ,         2  tan  (b 

JNowas  tan  26:= — ■ — ■ — x_ 

^      l-tan-(^ 

the  above  expression  becomes 

^;«  =  -^  X  ^-^^  (l  +  tan=  0) 
A  2  ?l  —  1  V 

which,  when  (f>  is  small  enough  to  make  tan-  <^  negligible  in  comparison  to  unity, 
reduces  to 

AX  a  —      n  Gv  cos  <^ 


or 


X         3/1-1  T 

X  cos  4>  rs   * 


AXo=^^X^^-^G.*  (213) 


248.  As  an  illustration,  let  us  return  to  our  standard  problem  12"  gun,  and 
compute  for  a  range  of  10,000  yards  the  data  contained  in  Column  14  of  the  range 
table;  change  of  range  for  motion  of  gun  in  plane  of  fire  of  12  knots.  We  have  the 
above  formula  and 

F-sin2<i  ^         12x6080         -,  . 

n  =  ^^-^  G.  =  ^Q^^Q^3  yards  per  second 

F  =  2900    log  3.46239 2  log  6.92478 

2<^  =  8°  26'  28"    sin  9.166G9-10 

^  =  32.2    log  1.50785 colog  8.49214-10 

X  =  30000    loff  4.47712 colog  5.52288-10 


n  =  1.278    log  0.10649 

2n  =  2.556 

2n-l  =  1.556    log  0.19200 colog  9.80800-10 

n  =  1.278 log  0.10649 

Z  =  30000    log  4.47712 

<j>  =  4:°  13'  14"    cos  9.99882-10 

F  =  2900    log  3.46239 colog  6.53761-10 

12    log  1.07918 

6080 log  3.78390 

60x60x3  =  10800    loi?  4.03342 colog  5.96658-10 


A7?,,rv  =  57  yards log  1.75770 

Lateral  249.  Now  let  Gz  be  the  resolved  part  of  the  motion  of  the  gun  at  right  anodes  to 

deviation  .  ^  '"     .         '"       . 

due  to  mo-    the  line  of  fire  in  foot-seconds.    Then,  in  addition  to  the  initial  velocity  V  in  the  line 

tion  of  gun. 

of  the  axis  of  the  gun,  the  projectile  on  leaving  the  gun  has  a  lateral  velocity  Gz,  and 
so,  as  may  be  seen  from  Figure  21(&),  the  real  plane  of  departure  makes  an  angle 

with  the  vertical  plane  of  the  gun's  axis  given  by  tan  y=  .  ^^ —  ,  and  the  resultant 
deviation  at  range  X  is  given  by 

Z)G  =  Xtany     or     Dc=  ^,  ^  ,   G,  (214) 

K  cos  <j!> 

*  The  above  is  the  formula  actually  employed.  There  seems  to  be  no  good  reason, 
however,  for  neglecting  the  tan^  tp,  for  tan-  4,  +  1  =  sec^  (p,  and  if  we  substitute  this  value, 
instead  of  dropping  the  tan^  0,  wa  would  get  as  the  final  result 

■^o  =  2n  — 1  X  Vcos^  ^  ^-^ 

which  is  equally  easy  for  work,  and  more  in  keeping  with  the  form  of  the  expression  for 
determining  the  deflection  due  to  motion  of  the  gun  given  in  equation  (214).  The  differ- 
ence is  not  material  however. 


VARIATION  OF  THE  TRAJECTORY  FROM  A  PLANE  CURVE     163 

250.  Taking  our   standard  problem   again,  we  have   the  above  formula  and 

^        13x6080         1  1 

G~=  „^    \^ — 7i  yards  per  secoiul. 
~      60x60x3-^  ^ 

Z  =  30000    log  4.47712 

7  =  3900    log  3.46240 eolog  6.53760-10 

(/>  =  4°   13'  14"    sec  0.00118 

12    log  1.07918 

6080    log  3.78390 

60x60x3  =  10800   log  4.03342 colog  5.9665S-10 

D,.o=^0.1  yards   log  1.84556 

Section  3. — The  Effect  of  the  Motion  of  the  Target  Upon  the  Motion  of  the  Projectile 

Relative  to  the  Target. 

251.  Motion  of  the  target  evidently  has  no  effect  upon  the  actual  flight  of  the 
projectile,  but  it  is  equally  clear  that  it  will  affect  the  relative  positions  of  the  target 
and  of  the  point  of  fall  of  the  projectile,  as  the  target  has  been  in  motion  during  the 
time  of  flight  of  the  projectile. 

252.  Evidently,  if  the  target  be  moving  in  the  line  of  fire  with  the  velocity  Tx,   Effect  of 
in  order  to  hit  it  the  sight  must  be  set  for  a  range  greater  or  less  than  the  true  range   target, 
at  the  instant  of  firing  by  the  distance  which  the  target  will  traverse  in  the  time  of 

flight,  or  T^T.  So,  also,  if  the  speed  of  the  target  at  right  angles  to  the  plane  of  fire 
be  Tz,  the  shot  will  fall  TzT  to  one  side  of  the  target  unless  that  much  deviation  is 
allowed  for  in  pointing.  Once  more  we  consider  the  motion  as  resolved  into  two 
components,  one  in  and  the  other  normal  to  the  plane  of  fire,  and  consider  the  two 
as  producing  results  entirely  independent  of  each  other.  And  it  is  readily  seen  that, 
for  the  effect  of  the  motion  of  the  target  we  must  correct  the  range  and  deviation  by 
the  quantities  given  by  the  expressions 

AXt  =  TxT  (215) 

Dt  =  T,T  (216) 

253.  For  our  standard  12"  gun,  again,  for  10,000  yards,  to  compute  the  data  in 
Columns  15  and  18  of  the  range  tables,  for  12  knots  speed  of  target,  the  work  would  be 

r=  12.43    log  1.09452 

12    log  1.07918 

6080    log  3.78390 

60  x  60  X  3  =  10800    log  4.03342 colog  5.96658  - 10 


R,._T  =  D^.T  =  84:  yards  log  1.92418 

Section  4- — The  Effect  Upon  the  Motion  of  the  Projectile  of  All  TJiree  Motions 

Combined. 

254.  In  the  preceding  sections  we  nave  discussed  the  effects  upon  the  motion  of 
the  projectile  of  the  wind  and  of  the  motions  of  the  firing  and  target  ships.  The 
resultant  combined  effect  of  all  three  of  these  causes  of  error  would  of  course  be 
obtained  by  computing  them  separately  and  then  performing  the  necessary  algebraic 


164 


EXTEEIOR  BALLISTICS 


additions,  first  for  all  range  effects  to  get  the  total  effect  upon  the  range,  and  then  of 
all  deflection  effects  to  get  the  total  effect  in  deflection. 

Note. — Professor  Alger  appends  to  this  chapter  the  following  foot-note: 
The  method  herein  adopted  for  the  treatment  of  the  problem  of  wind  effect  was  first 
set  forth,  so  far  as  I  am  aware,  in  General  Didion's  Traite  de  Ballistique  though  it  has 
been  generally  accepted  since.  It  is  mathematically  correct  only  for  spherical  projectiles, 
to  the  motion  of  which  the  air  offers  a  resistance  which  is  independent  of  the  direction  of 
motion.  With  elongated  projectiles  it  will  be  seen  that  the  initial  motion  relative  to  the 
air  is  not  exactly  in  the  line  of  the  projectile's  axis,  so  that  we  have  no  right  to  assume, 
as  we  do,  that  the  flight  relative  to  the  air  is  the  same  when  the  air  is  moving  as  when  it  is 
still.  It  has  been  supposed  by  some  writers  that  the  lateral  wind  component  produces  the 
same  pressure  on  the  side  of  the  moving  projectile  as  it  would  if  the  projectile  were  sta- 
tionary, and  that  the  deviation  can  be  computed  upon  that  basis.  If  this  were  true,  the 
deviation  would  be  proportional  to  the  square  of  the  lateral  wind  component,  whereas  it  is 
really  much  more  nearly  proportional  to  its  first  power.  Actually  the  pressure  is  much 
greater  when  the  projectile  is  moving  at  right  angles  to  the  wind  current  than  when  it  is 
stationary,  on  account  of  the  increased  number  of  air  particles  which  strike  it. 


EXAMPLES. 

1.  Compute  the  errors  in  range  and  in  deflection  caused  by  the  wind  components 
as  "iven  below. 


DATA. 

ANSWERS. 

Projectile. 

Wind  component  in  k 
per  hour. 

nets 

V. 

f.s. 

Range. 
Yds. 

In  line 

of  fire. 

Perpen 
to  line 

dicular 

In  line 

of  fire. 

Perpen 
to  line 

iicular 
of  fire. 

of  fire. 

H 

d. 
In. 

10. 

Lbs. 

c. 

o 

Knots. 

With  or 
against. 

Knots. 

To  the 

right 

or  left 

Yards. 

Short 
or  over. 

Yards. 

To  the 

right 

or  left. 

A.. 

3 

13 

1.00 

1150 

3000 

8 

With 

6 

Right 

16.5 

Over 

0.3 

Kight 

B.. 

3 

13 

1.00 

2700 

4500 

10 

Against 

8 

Left 

34.5 

Short 

19.1 

Left 

C. 

4 

33 

0.67 

2900 

4000 

11 

With 

9 

Right 

12.7 

Over 

5.7 

Right 

D. 

5 

50 

1. 00 

3150 

4500 

13 

Against 

11 

Left 

26 . 8 

Short 

14.4 

Left 

E.. 

5 

50 

0.61 

3150 

4500 

14 

With 

13 

Right 

17.1 

Over 

S.S 

Right 

F.. 

6 

105 

0.61 

2600 

13600 

15 

Against 

14 

Left 

146.9 

Short 

88.8 

Left 

G.. 

6 

105 

1. 00 

2800 

4500 

16 

\\itli 

15 

Right 

25 . 9 

Over 

14.0 

Right 

H. 

6 

105 

0.01 

2800 

4000 

17 

Against 

16 

Left 

13.3 

Short 

6.4 

Left 

I.. 

7 

165 

1.00 

2700 

7500 

18 

With 

17 

Right 

74.2 

Over 

44.3 

Right 

J.. 

7 

165 

O.OI 

2700 

7500 

19 

Against 

18 

Left 

47.0 

Short 

24.8 

Left 

K. 

8 

260!0.6I 

2750 

8500 

20 

With 

10 

Right 

51.5 

Over 

26.9 

Right 

L.. 

10 

510  1. 00^2700 

10500 

10 

Against 

20 

Left 

100.4 

Short 

64.7 

Left 

M. 

10 

510  0.61 

2700 

11500 

18 

Witli 

19 

Right 

69.0 

Over 

40.2 

Right 

N. 

12 

870  0.61 

2000 

10500 

17 

With 

18 

Lelt 

141.5 

Over 

87.0 

Left 

0.. 

13 

iisoli.oo 

2000 

10500 

16 

Against 

17 

Right 

93.3 

Short 

54.8 

Right 

P.. 

13 

ll;W,0.74 

2000 

11500 

15 

With 

16 

Left 

79.6 

Over 

44.4 

Left 

Q.. 

14 

1400  0.70 

2000 

14500 

14 

Against 

15 

Pvight 

105.6 

Short 

58 . 1 

Right 

R.. 

14 

14000.70 

2600 

14000 

13 

With 

14 

Left 

03.7 

Over 

37.3 

Left 

VAEIATION  OF  THE  TEAJECTORY  FEOM  A  PLANE  CUEVE     161 


2.  Compute  the  errors  in  range  and  in  deflection  caused  by  the  motion  of  the 
gun  as  given  below.     Conditions  standard. 


DATA. 

ANSWERS. 

Pr 

ojeetile. 

Speed  component  in  knots 
per  hour. 

r. 

f.s. 

Range. 
Yds. 

In  line 

of  fire. 

Perpen 
to  line 

dicular 

In  line 

of  fire. 

Perpen 
to  line 

dicular 
of  fire. 

of  fire. 

s 

d. 
In. 

Lbs. 

c. 

1 

Knots. 

With  or 
against. 

Knots. 

To  the 
right 
or  left. 

Yards. 

Short 
or  over. 

Yards. 

To  the 

right 

or  left. 

A.. 

3 

13 

1.00 

1150 

2000 

6 

Against 

8 

Left 

14.7 

Short 

23.6 

Left 

B.. 

3 

13 

1.00,2700 

3500 

7 

With 

8 

Right 

10.2 

Over 

17.5 

Right 

C. 

4 

33 

0.67  21)00 

3000 

8 

Against 

9 

Left 

11.5 

Short 

15.7 

Left 

D. 

5 

50 

1. 0013150 

3500 

9 

With 

10 

Right 

12.4 

Over 

18.8 

Right 

E.. 

5 

50 

0.61j3150 

3800 

10 

Against 

11 

Left 

16.4 

Short 

22.4 

Left 

F.. 

6 

105 

0.6l'2600 

12600 

11 

With 

13 

Left 

58.3 

Over 

108.4 

Left 

G.. 

6 

105 

1 .  00  2800 

4000 

13 

Against 

14 

Right 

24.1 

Short 

33.8 

Right 

H. 

G 

105 

O.Gl  2800 

3000 

14 

With 

15 

Left 

22.2 

Over 

27.1 

Left 

I.. 

7 

165 

1.00  2700 

6500 

15 

Against 

16 

Right 

43.1 

Short 

65.2 

Right 

J.. 

7 

165 

0.61  2700 

6700 

16 

With 

17 

Left 

52.8 

Over 

71.4 

Left 

K. 

8 

260 

0.61,2750 

7500 

17 

With 

18 

Right 

62.6 

Over 

83.1 

Right 

L.. 

10 

510,1.00  2700 

9500 

18 

Against 

19 

Left 

76.3 

Short 

113.5 

Left 

M. 

10 

510 

0.612700 

10500 

19 

With 

20 

Right 

97.6 

Over 

132.0 

Right 

N. 

12 

870 

0.61  2000 

23000 

20 

Against 

19 

Left 

182.3 

Short 

262.0 

Left 

0.. 

13 

1130 

1 .  00  2000 

10000 

19 

With 

18 

Right 

118.0 

Over 

154.7 

Right 

P.. 

13 

1130 

0. 7412000 

11000 

18 

Against 

17 

Left 

128.4 

Short 

160.8 

Left 

Q.. 

14 

1400 

0.7012000 

14000 

17 

With 

16 

Right 

149.5 

Over 

195.4 

Right 

R.. 

14 

1400 

0.70,2600 

13700 

16 

Against 

15 

Left 

109.4 

Short 

134.8 

Left 

3,  Compute  the  errors  in  range  and  in  deflection  caused  by  the  motion  of  the 
target  as  given  below.    Conditions  standard. 


DATA. 

ANSWERS. 

Projectile. 

Speed  component  in  k 
per  hour. 

nots 

In  line  of  fire. 

Perpendicular 

to  line  of  fire. 

V. 

f.s. 

Range. 
Yds. 

In  line 

of  fire.- 

Perpen 
to  line 

dicular 
of  fire. 

B 

d. 
In. 

w. 
Lbs. 

c. 

0) 

3 
o 

Knots. 

With  or 
against. 

Knots. 

To  the 

right 

or  left. 

Yards. 

Short 
or  over. 

Y'ards. 

To  the 

right 

or  left. 

A.. 

3 

13 

1.00 

1150 

1800 

7 

With 

8 

Right 

21.4 

Short 

24.4 

Left 

B.. 

3 

13 

1.00 

2700 

3300 

8 

Against 

9 

Left 

26.2 

Over 

29.4 

Right 

C. 

4 

33 

0.67 

2900 

2800 

9 

With 

10 

Right 

17.3 

Short 

19.2 

Left 

D. 

5 

50 

1.00  3150 

3300 

10 

Against 

11 

Left 

24.0 

Over 

26.4 

Right 

E.. 

5 

50 

0.613150 

3400 

11 

With 

13 

Right 

24.1 

Short 

28.5 

Left 

b\. 

6 

105 

0.612000 

13800 

13 

With 

14 

Left 

204.3 

Short 

220.1 

Right 

a.. 

6 

105 

1.00  2800 

3800 

14 

Airainst 

15 

Right 

41.1 

Over 

44.0 

Left 

H. 

6 

105 

0.612800 

2800 

15 

With 

16 

Left 

28.1 

Short 

30.0 

Right 

1.. 

7 

105 

1.00  2700 

6300 

16 

Against 

17 

Right 

91.1 

Over 

96.9 

Left 

J.. 

7 

165 

0.612700 

6700 

17 

Witli 

18 

Left 

89.5 

Short 

94.7     Right 

K. 

8 

260 

0.612750 

7300 

18 

Against 

19 

Left 

99.2 

Over 

104.6 

Right 

L.. 

10 

510 

1.00  2700 

9300 

19 

With 

20 

Right 

157.9 

Short 

166.2 

Left 

M. 

10    510 

0.6l'2700 

10300 

20 

Against 

19 

Left 

162.7 

Over 

154.6 

Right 

N  . 

12    870 

0.61 12900 

20200 

19 

With 

20 

Right 

326.7 

Short 

343.9 

Left 

0.. 

13  1130 

1.00  2000 

9300 

18 

Against 

19 

Left 

188.6 

Over 

199.1  !  Right 

P.. 

131130 

0.74  2000 

10300 

17 

With 

18 

Right 

187.6 

Short 

198.6  1  Left 

Q.. 

14  1400 

0.70  2000 

13300 

16 

Against 

17 

Left 

236.8 

Over 

251.6     Right 

R.. 

14  1400 

0.70  2600 

13700 

15 

With 

16 

Right 

172.4 

Short 

184.5     Left 

CHAPTER  15. 

DETERMINATION  OF  JUMP.    EXPERIMENTAL  RANGING  AND  THE 
REDUCTION  OF  OBSERVED  RANGES. 

Jump.  255.  Primarily  and  in  its  narrowest  sense,  jump  is  the  increase  (algebraic,  and 

generall}^  positive)  in  the  angle  of  elevation  resulting  from  the  angular  motion  of  the 
gun  in  the  vertical  plane  caused  by  the  shock  of  discharge,  as  a  result  of  which  the 
projectile  strikes  above  (for  positive  jump;  below  for  negative)  the  point  at  which 
it  theoretically  should  for  the  given  angle  of  elevation.  A  definition  which  thus  con- 
fines jump  to  the  result  of  such  angular  motion  is  a  narrow  and  restricted  one,  how- 
ever, and  other  elements  may  enter  to  give  similar  results,  all  of  which  may  be  and 
are  properly  included  in  that  resultant  variation  generally  called  jump.  For  instance, 
in  the  old  gravity  return  mounts,  the  gun  did  not  recoil  directly  in  the  line  of  its  own 
axis,  as  it  does  in  the  most  modern  mounts,  but  rose  up  an  inclined  plane  as  it 
recoiled.  As  the  projectile  did  not  clear  the  inuzzle  until  the  gun  had  recoiled  an 
appreciable  distance,  this  upward  motion  of  the  gun  imparted  a  similar  upward 
motion  to  the  projectile,  which  resulted  in  making  the  projectile  strike  slightly 
higher  than  it  would  otherwise  have  done.  This  small  discrepancy,  unimportant  at 
battle  ranges,  but  necessarily  considered  in  such  work  as  firing  test  shots  at  armor 
plate  at  close  range,  was  properly  included  in  the  jump.  Also  most  modern  guns  of 
any  considerable  length  have  what  is  known  as  "  droop,"  that  is,  the  muzzle  of  the 
gun  sags  a  little,  due  to  the  length  and  weight  of  the  gun,  the  axis  of  the  gun  being  no 
longer  a  theoretical  straight  line;  and  this  causes  the  projectile  to  strike  slightly 
lower  than  it  otherwise  would,  and  introduces  another  slight  error  which  may 
properly  be  included  in  the  jump.  Also  it  is  probable  thai  this  droop  causes  the 
muzzle  of  the  gun  to  move  slightly  in  firing  as  the  gun  tends  to  straighten  out  under 
internal  pressure,  and  perhaps  this  motion  tends  to  produce  another  variation, 
"  whip,"  in  the  motion  of  the  projectile,  which  would  modify  the  result  of  the  droop. 
All  these  may  therefore  be  properly  included  in  the  jump. 

256.  This  matter  has  a  direct  bearing,  under  our  present  system  of  considering 
such  matters,  upon  the  factor  of  the  ballistic  coefficient  which  we  have  designated  as 
the  coefficient  of  form  of  the  projectile,  and  which  is  supposed,  under  our  previous 
definition,  to  be  the  ratio  of  the  resistance  the  projectile  meets  in  flight  to  the  resist- 
ance that  would  be  encountered  in  the  same  air,  at  the  same  velocity,  by  the  standard 
projectile;  that  is,  by  a  projectile  about  three  calibers  long  and  similar  in  all  respects 
except  in  possessing  a  standard  head,  namely,  one  whose  ogival  has  a  two-caliber 
radius.  Imagine  that  the  gun  jumps  a  little  and  increases  the  range  in  so  doing. 
It  gives  the  same  range  as  a  similar  gun  firing  without  jump  a  projectile  exactly 
similar  in  all  respects  except  in  possessing  a  slightly  lower  coefficient  of  form.  Sup- 
pose a  gun  droops  and  shoots  lower.  The  coefficient  of  form  calculated  back  from  the 
range  obtained  by  actual  firing  would  work  out  a  little  large.  And  in  practice  we 
would  proba])ly  have  both  jump  and  droop  afi'ecting  the  range,  but  by  our  method  of 
determining  the  coefficient  of  form  from  actual  firing,  by  comparing  actual  with 
computed  ranges,  all  such  effects  are  hidden  in  the  found  value  of  the  coefficient  of 
form. 
Broader  defi-  257.  As  a  matter  of  fact,  as  intimated  above,  the  value  of  the  coefficient  of  form 

coefficient    is  determined  by  firing  ranging  shots,  and  then  computing  its  value  from  the  results. 

of  form.  ^ 


VARIATION"  OF  THE  TRAJECTORY  FROM  A  PLANE  CURVE     1G7 

This  coefficient  of  form  therefore  includes  not  only  the  results  of  variations  in  form 
of  projectiles,  etc.,  but  also  variations  in  range  resulting  from  jump,  droop,  whip,  etc., 
In  fact,  in  the  sense  in  which  we  now  use  the  term,  coefficient  of  form  might  better 
be  defined  as  "  that  value  which,  if  substituted  for  c  in  the  usual  formula?,  will,  for 
the  given  elevation,  velocity,  weight  of  projectile,  etc.,  make  the  computed  range 
come  out  in  agreement  with  that  actually  attained  in  firing,  after  making  all  correc- 
tions to  the  firing  results  for  atmospheric  density,  etc." 

258.  Thus  a  person  looking  over  the  range  table  computations  for  the  first  time 
would  say  off  hand  tliat  jump,  droop,  etc.,  were  neglected.  Closer  study,  however, 
makes  it  evident  that  the  adopted  procedure  amounts  to  taking  jump,  etc.,  into 
consideration  as  actually  found  to  exist;  not  in  assuming  that  it  is  zero,  and,  in  fact, 
not  greatly  concerning  ourselves  as  to  just  what  its  value  really  is  (as  we  know  it  to  be 
small),  but  still  following  a  method  that,  for  a  given  jump,  etc.,  gives  a  correct  com- 
puted result  at  a  given  range,  and  which  checks  well  at  all  other  ranges.  If  it  be 
objected  that  different  guns  of  the  same  type  may  jump,  etc.,  differently,  it  may  be 
answered  that  the  coefficient  of  form  used  is  an  average  of  the  values  obtained  in  a 
great  number  of  firings  of  different  individual  guns  of  the  same  type,  and  is  really 
preferable  to  that  obtaiiied  by  a  complete  ranging  of  a  single  gun.  As  a  matter  of 
fact,  variations  in  the  value  of  the  coefficient  of  form  obtained  do  not  seem  to  go  with 
certain  individual  guns  more  than  with  others,  so  the  range  tables  are  equally  good 
for  all  guns  of  the  type.  In  other  words,  this  method  of  procedure  produces  results 
that  are  within  the  limits  of  accuracy  obtainable,  and  any  errors  that  follow  its  use 
must  necessarily  be  included  in  those  inherent  errors  of  the  gun  which  must  always 
exist,  and  which,  after  all  possible  precautions  have  been  tt:ken,  will  inevitably  make 
it  impossible  to  have  all  the  shot  from  the  same  gun,  when  fired  under  exactly  similar 
conditions,  always  strike  in  exactly  the  same  spot. 

259.  Having  explained  how  jump,  etc.,  even  if  it  exists,  is  looked  out  for  by  our 
methods,  we  may  now  go  on  and  state  that,  as  a  matter  of  fact,  it  does  not  seem  to 
exist  to  any  extent  appreciable  in  the  service  use  of  guns,  and  it  may  therefore  be  said 
that  it  is  a  matter  which  does  not  concern  an  officer  afloat.  He  should,  however,  have 
a  knowledge  of  the  principles  laid  down  in  this  chapter,  in  order  that  he  may  recog- 
nize unusual  and  abnormal  conditions  should  they  be  found  to  exist  under  special 
conditions. 

260.  To  determine  the  value  of  the  coefficient  of  form  for  a  projectile  for  our  practical 
standard  problem  12"  gun,  the  gun  was  mounted  at  the  Proving  Ground,  and  laid  at   tion  of  value 
an  angle  of  elevation  of  8°,  using  a  gunner's  quadrant.     Correction  for  height  of   of  form. 
trunnions  above  the  water  level,  for  sphericity  of  the  earth,  etc.,  makes  this  angle  the 
equivalent  of  an  angle  of  elevation  of  8°  04'.    The  gun,  when  fired  at  this  elevation, 

at  2900  f.  s.  initial  velocity,  should  range  about  16,140  yards,  from  the  range  talkie, 
when  the  observed  fall  has  been  reduced  to  standard  atmospheric  conditions,  but  of 
course  this  is  not  perfectly  obtained.  Say  the  projectile  falls  100  yards  short  of  the 
computed  range.  It  then  remains  to  determine  the  value  of  the  coefficient  of  form 
which  produced  this  variation,  and  this  may  be  done  by  calculating  back  by  the 
methods  that  have  already  been  explained  in  this  book.  In  practice,  at  the  Proving 
Ground,  however,  in  order  to  avoid  the  constant  repetition  of  tedious  computations, 
the  method  actually  employed  is  to  work  out  a  few  ranges  for  different  values  of  the 
coefficient  of  form,  and  to  make  a  curve  for  the  results.  The  curve  is  made  for  argu- 
ments "  coefficient  of  form  "  and  "  error  from  corrected  range  table  range,"  and  one 
curve  is  needed  for  each  caliber  and  service  velocity.  Having  such  curves  and  the 
results  of  ranging  shots,  it  is  a  quick  and  easy  matter  to  take  from  the  proper  curve 
the  value  of  the  coefficient  of  form  for  each  projectile.    These  values  are  tabulated, 


168 


EXTEEIOR  BALLISTICS 


Computation 

of  range 

tables. 


Keduction 

of  observed 

ranges. 


and  a  running  record  is  kept,  so  that  a  great  number  of  results  will  be  available  as  a 
cumulative  check  on  the  range  table.  For  a  new  caliber,  a  curve  of  "corrected 
range ''  and  "  coefficient  of  form  "  is  kept  until  enough  data  has  been  collected  with 
which  to  start  a  range  table.  For  rough  work,  the  formula  for  change  of  range 
resulting  from  a  variation  in  the  value  of  the  ballistic  coefficient  may  be  used  in  the 
absence  of  curves. 

261.  Prior  to  the  appearance  of  the  present  range  tables,  guns  were  ranged  by 
firing  experimental  shots  at  a  number  of  different  angles  of  elevation,  and  a  curve  of 
angles  and  ranges  was  plotted.  From  these  faired  curves  the  angles  corresponding 
to  all  ranges  were  taken  and  a  range  table  was  made  up  from  the  results.  As  more 
confidence  in  the  mathematical  process  was  acquired,  through  the  accumulation  of 
considerable  data,  we  began  to  get  our  range  tables  by  computation,  gradually 
abandoning  the  old  system  of  ranging  by  experimental  firing;  and  the  use  of  the  value 
of  the  coefficient  of  form  as  unity,  with  the  projectiles  then  in  use,  was  found  to  give 
range  tables  that  agreed  with  the  results  of  experimental  ranging.  When  different 
lots  of  projectiles  are  presented  for  acceptance,  a  few  have  to  be  tested  for  flight  from 
each  lot;  and  these  are  ranged  at  the  Proving  Ground  at  8°  elevation  in  all  cases,  in 
order  to  make  comparisons  possible.  At  this  angle  there  are  no  dangerous  ricochets, 
and  variations  in  the  coefficient  of  form  and  differences  between  different  projectiles 
will  show  up  best  at  these  long  ranges.  With  a  coefficient  of  form  accurately  de- 
termined by  firing  at  the  longest  practicable  ranges,  we  can  compute  an  extremely 
accurate  range  table  extending  down  through  the  medium  and  short  ranges.  The 
method  of  ranging  only  at  a  single  elevation  was  therefore  adopted,  an  occasional 
check  by  firing  at  short  ranges  being  made. 

2S2.  The  process  of  experimental  ranging,  as  formerly  carried  out,  was  to  fire 
a  number  of  shots  at  different  angles  of  elevation.  The  results  for  these  shots  were 
reduced  to  standard  conditions,  and  the  reduced  observed  ranges  were  plotted  as 
abscissae,  with  the  corresponding  angles  of  elevation  as  ordinates.  A  fair  curve  was 
then  drawn  through  these  points,  and  from  this  curve  the  angle  of  elevation  corre- 
sponding to  any  range  could  be  obtained. 

263.  The  process  of  reducing  observed  ranges  to  standard  conditions  was 
carried  out  in  accordance  with  the  principles  and  formulae  already  explained  in  this 
book,  and  this  still  has  to  be  done  for  every  ranging  shot  fired ;  but  as  this  process  is 
one  that  pertains  purely  to  Proving  Ground  work  and  has  no  bearing  on  the  service 
use  of  the  gun,  it  is  not  considered  necessary  to  go  into  it  at  length  here ;  nor  is  it 
considered  necessary  to  further  discuss  the  details  of  the  methods  used  for  deter- 
mining the  actual  magnitude  of  the  jump,  etc. 


PAET  IV. 
RAXGE  TABLES;  THEIR  COMPUTATION  AND  USE. 

IXTRODUCTIOX  TO  PART  IV. 

Having  completed  the  study  of  all  computations  connected  with  the  trajector}^  in 
air,  both  as  a  plane  curve  and  allowing  for  existing  variations  from  that  plane,  we  are 
now  in  a  position  to  make  use  of  our  knowledge  in  a  practical  way.  The  practical 
and  useful  expression  of  the  knowledge  thus  acquired  takes  the  form  of:  first,  the 
preparation  of  the  range  tables;  and  after  that,  second,  their  use.  Part  IV  will  be 
devoted  to  a  consideration  of  the  range  tables  from  these  two  points  of  view :  first,  as 
to  their  preparation;  and,  second,  as  to  their  actual  practical  use  in  service.  Each 
column  in  the  tables  will  be  considered  separately,  the  method  and  computations  by 
which  the  data  contained  in  it  is  obtained  will  be  indicated,  and  then  consideration 
will  be  given  to  the  practical  use  of  this  data  by  officers  aboard  ship  in  service. 


CHAPTER  16. 

THE  COMPUTATION  OF  THE  DATA  CONTAINED  IN  THE  RANGE  TABLES  IN 

GENERAL;  AND  THE  COMPUTATION  OF  THE  DATA  CONTAINED  IN 

COLUMN  9  OF  THE  RANGE  TABLES. 

New  Symbols  Introduced. 

E^.  . .  .Penetration,  in  inches,  of  Harvcyized  armor  by  capped  projectiles. 
E.,.  . .  . Penetration,  in  inches,  of  face-hardened  armor  by  capped  projectiles. 
K.  . .  .Constant  factor  for  face-hardened  armor. 
K' .  . .  .Constant  factor  for  Harveyized  armor. 

264.  With  the  single  exception  noted  in  the  next  paragraph,  we  have  now  con- 
sidered in  detail  the  formulge  and  methods  by  which  the  data  in  each  of  the  columns 
of  the  range  table  is  computed.    Summarized,  this  is  as  follows : 

^  ,  Ti  rr  1  1  Chapter  in  tliis 

,■ Column  m  Range  Table ^  ,  .        ,•  ■ 

XT  T\  i.     n     t-  ^      -i  book  in  which 

rso.  Data  Contained  ,    •      , 

explained 

1.  . .  .Range.    This  is  the  foundation  column  for  which  the  data  in  the    "I    No  explanation 

other  columns  is  computed.     There  are  therefore  no  j         necessary. 

computations  in  regard  to  it. 

2 . . . .  Angle  of  departure 8 

3 Angle    of   fall 8 

4 Time  of   flight 8 

5 . . .  .  Striking  velocity    ii 

6 Drift    13 

7 . . .  .  Danger  space  for  a  target  20  feet  high 5 

8 . . .  .  Maximum   ordinate    8  and  9 

9.  . .  .Penetration   of  armor This  chapter 

10. . .  .Change  of  range  for  variation  of  ±  50  foot-seconds  initial  velocity 12 

11.  . .  .Change  of  range  for  variation  of  ±  dw  pounds  in  weight  of  projectile 12 

12. . .  .Change  in  range  for  variation  of  ±  10  per  cent  in  density  of  air 12 

13. . .  .Change  in  range  for  wind  component  in  plane  of  fire  of  12  knots 14 

14.  . .  .Change  in  range  for  motion  of  gun  in  plane  of  fire  of  12  knots 14 

15. . .  .Change  in  range  for  motion  of  target  in  plane  of  fire  of  12  knots 14 

16 ...  .  Deviation  for  lateral  wind  component  of  12  knots 14 

17.  . .  .Deviation  for  lateral  motion  of  gun  perpendicular  to  line  of  fire;  speed  12  knots.    14 

18.  ..  .Deviation  for  lateral  motion  of  target  perpendicular  to  line  of  fire;    speed  12 

knots     14 

19.  . .  .Change  of  height  of  impact  for  variation  of  ±  100  yards  in  sight  bar 12 

265.  The  subject  of  penetration  of  armor  is  one  which  does  not  properly  Ijelong 
to  the  subject  of  exterior  ballistics,  but  this  text  book  is  compiled  from  the  special 
point  of  view  of  the  computation  and  use  of  the  range  tables,  and  as  Column  9  of  each 
of  these  tables  gives  the  penetration,  the  subject  is  covered  here  in  a  brief  way,  in 
order  that  the  full  range  table  computations  may  be  covered  together. 

266.  In  the  earlier  range  tables,  the  penetration  of  armor  was  given  for  Harvey-  penetration 
ized  armor,  and  formulas  devised  by  Commander  Cleland  Davis,  U.  S.  jST.,  were   *°*'™"^*' 
employed  in  the  computation.    For  later  armor,  the  range  tables  give  the  penetration 

of  face-hardened  armor  by  capped  projectiles,  this  data  being  computed  by  the  use 
of  De  Marre's  formula.  The  heading  of  the  column  in  each  range  table  shows  which 
type  of  armor  is  referred  to  in  that  particular  table.  Given  the  penetration  in 
Harveyized  armor,  the  penetration  for  face-hardened  armor  may  be  approximately 
obtained  by  multiplying  the  known  figure  for  Harveyized  armor  by  0.8.  Davis's 
formulre  for  Harveyized  armor  are : 


17S 


EXTERIOR  BALLISTICS 


in  which 


For  projectiles  without  caps. 

^0.5^0.75 

E  =  i]-\e  penetration  of  Harvey ized  armor  in  inches. 


or 


^0.75  ■_   VW" 


(217) 


t'  =  the  striking  velocity  in  foot-seconds. 
U'=:the  weight  of  the  projectile  in  pounds. 
fZ=:the  diameter  of  the  projectile  in  inches. 
lo^  A'  =  3.34512. 


For  capped  projectiles. 


^=      .,,0.5    -^^ 


VIV 


,0.5 


or     ^o.s_jii^ 


(218) 


in  which  log  Z'=  3.25312,  and  the  other  quantities  are  as  before.    De  Marre's  formula 
for  face-hardened  armor  is 

^0.75^0. 


K     or     E'>-'  = 


VlV 


(219) 


in  which  log  Z=  3.00945,  £'  =  the  penetration  of  oil  tempered  and  annealed  armor 
that  has  not  been  face  hardened.  For  face-hardened  armor  (for  the  range  tables 
accompanying  this  book  and  marked  as  C,  F,  H,  K,  M,  N,  P,  Q  and  Pk,),  the  results 
obtained  by  the  use  of  the  above  formula  must  be  divided  by  a  divisor  known  as 
De  Marre's  coefficient,  which  has  been  found  to  be  1.5  for  such  purpose.  (For  the 
other  range  tables  accompanying  this  book,  the  value  of  this  coefficient  was  taken  as 
unity.) 

267.  As  an  example  of  the  work  under  Davis's  formula,  let  us  compute  the 
penetration  by  a  capped  projectile  of  Harveyized  armor  by  the  5"  gun;  F  =  3150  f.  s., 
w  =  h()  pounds;  for  a  range  of  4000  yards,  first  for  a  projectile  for  which  c  —  l,  and 
next  for  a  projectile  whose  coefficient  of  form  is  0.61.  For  these  two  projectiles,  the 
range  tables  give  the  remaining  velocities  at  the  given  range  as  v-l  =  1510  f.  s.  for 
c=1.00  and  1^2  =  2098  f.  s.  for  c  =  0.61. 

w=50    log  1.69897 0.5  log  0.84948 

K'=    log  3.25312 colog  6.74688-10 

d  =  5    log  0.69897 0.5  log  0.34948 0.5  colog  9.65052-10 


10 


,loff  7.246S8-10 


log  7.24688- 

^.^  =  1510    log  3.17898 

t;.  =  2098    log  3.32181 


,&o.8    0.8  log  0.42586 0.8  log  0.56869 

J5^i  =  3.4067"    log  0.53233 

ii;2  =  5.1380"    log  0.71086 

Comparison  268.  As  the  Coefficient  of  form  does  not  enter  into  the  above  equation,  we  see 

short  pointed  that  the  Only  thing  that  gives  a  long  pointed  projectile  a  greater  penetration  than 
projec  I  es.  ^  ]j|^jj-,^  pointed  one  at  the  same  range  is  the  fact  that,  at  that  range,  the  long  pointed 
projectile  will  have  a  greater  striking  velocity  than  the  blunt  one.  As  a  matter  of 
fact,  as  far  as  their  effect  upon  armor  plate  is  concerned,  the  two  projectiles  are  the 
same ;  for  the  main  body  of  the  projectile  is  the  same  in  each  case,  the  only  difference 
between  them  being  in  the  shape  of  the  wind  shield.  In  other  words,  that  part  of  the 
projectile  which  really  acts  to  penetrate  armor  is  the  same  for  both  the  standard  and 
tor  the  long  pomted  siiell,  but  one  has  no  wind  shield  and  the  other  a  sharply  pointed 
one,  the  actual  points  of  the  two  shells  being  equally  efficient  in  their  effect  upon  the 
penetration.  No  difference  in  penetration  could  therefore  be  expected  for  equal 
striking  velocities. 


EANGE  TABLES;  THEIR  COMPUTATION  AND  USE  173 

269.  Now  let  us  take  our  standard  problem  and  determine,  by  the  use  of 
De  IMarre's  formula,  the  penetration  of  face-hardened  armor  at  10,000  yards,  for  the 
12"  gun  of  the  problem.    The  range  table  gives  t'  =  2029  f.  s.  for  that  range. 

j^o--=  4Ci.   ill  which  log /i  =3.00945 

r  =  2029    log  3.3072S 

w  =  8;0    log  2.93952 0.5  log  1.46976 

K=  log  3.00945 colog  6.99055-10 

d  =  12   loff  1.07918.  .  .0.75  W  0.80938 0.75  colog  9.19062-10 


tables. 


E"-'   0.7  log  0.95821 

^2X1.5    log  1.36887 

1.5    log  0.17609 colog  9.S2391-10 

^,  =  15.59"    log  1.19278 

270.  Having  learned  how  to  compute  the  penetration  of  armor  by  a  given  pro-  practical 
jectile  at  a  given  range,  we  are  now  in  a  position  to  discuss  the  practical  methods  of  range  °° 
used  in  actually  making  the  computations  for  a  range  table.  The  labor  involved  is 
of  course  very  great,  so  much  care  has  been  taken  to  get  up  special  forms,  and  these 
are  printed  and  kept  on  hand  for  the  work.  These  forms  are  given  in  the  follow- 
ing pages,  the  figures  given  in  them  being  for  the  problems  that  we  have  already 
worked  out  item  by  item,  and  shown  here  as  they  would  appear  in  the  work  of  pre- 
paring the  12"  range  table  with  the  data  for  which,  at  a  single  range  only,  we  have 
been  working.  These  forms  therefore  show  only  the  computations  involved  for 
10,000  yards  range.  In  computing  the  actual  table,  the  work  is  done  first  for  1000 
yards,  then  for  1500  yards,  etc.,  for  each  500-yard  increment  in  range,  the  values 
between  the  computed  values  being  obtained  by  interpolations,  which  interpolations 
are  not  difficult,  as  in  most  cases  the  second  and  third  differences  are  negligible. 
As  the  range  table  of  the  particular  gun  in  question  runs  up  to  24,000  yards,  and 
as  computations  must  be  made  for  every  500  yards,  it  will  be  seen  that  the  work 
must  be  repeated  for  every  500  yards  from  1000  yards  up,  which  will  involve  47  com- 
plete computations  like  the  one  shown  in  the  following  forms.  This  will  give  some 
idea  of  the  magnitude  of  the  work  involved,  and  of  the  necessity  for  having  special 
forms  prepared,  and  of  otherwise  reducing  the  labor  and  increasing  the  accuracy  as 
much  as  possible.  Therefore,  if  much  of  this  kind  of  work  is  to  be  done,  forms  similar 
to  the  ones  shown  should  be  prepared  before  commencing  it  (if  a  supply  of  the  printed 
forms  be  not  at  hand)  ;  but  if  only  a  single  problem  is  to  be  worked,  as  will  generally 
be  the  case  in  the  instruction  of  midshipmen,  then  the  forms  given  in  the  previous 
chapters  of  this  book  should  be  used,  as  showing  more  clearly  the  nature  of  the 
problems  involved  and  the  methods  of  solving  them.  In  solving  problems  under  this 
chapter  the  forms  given  below  must  be  used. 

271.  The  form  given  for  determining  the  angle  of  departure  for  a  given  range 
provides  space  for  only  three  approximations;  if  more  approximations  are  necessary 
to  get  the  correct  result,  the  form  is  simply  extended. 

272.  It  is  to  be  noted  that,  in  order  to  get  smooth  curves  on  the  deflection  drums 
of  the  sights,  it  is  necessary  in  some  cases  to  fair  the  computed  drift  curve,  and  this 
produces,  in  some  places,  a  small  difference  between  the  computed  drift  and  that 
shown  in  the  range  tables.  Thus  the  computed  drift  for  our  standard  problem  was 
26.6  yards,  while  in  the  range  tables  it  is  given  as  26.8  yards. 

273.  The  problem  before  us  then  is  to  compute,  for  a  range  of  10,000  yards,  the 
data  for  the  columns  of  the  range  table  for  the  12"  gun  for  which  7  =  2900  foot- 
seconds,  «'  =  870  pounds,  and  c  =  0.61.      The  forms  and  work  follow. 


174  - 


EXTERIOR  BALLISTICS 


STANDARD  FORMS  FOR  COMPUTATION  OF  RANGE  TABLE  DATA. 

Specific  Problem. 

Compute  the  data  for  all  columns  of  the  range  table  for  a  range  of  10,000  yards  for  a 
12"  gun  for  which  V  =  2900  f.  s.,  w  =  870  pounds,  and  c  =  0.61. 

j  Form  No.  1. 

For  Computation  of  Angle  of  Departure. 

Column  2, 


Uncorrected  value  of  C 

Ci  =  ^„  =  9.9045 
cd- 


log   10  .  . 

colog  c. 
colog  d^ 


log  C, 


2 

9 

3 

9 

5 

2 

0 

2 

1 

4 

6 

7 

7 

8 

4 

1 

6 

4  1 

0 

9 

9 

5 

8 

3  i 

Z,  =  -^=  3U29 


log  X. . . 
colog  Ci 


log  Z, . 


4 

4 

7 

7 

1 

2 

9 

0 

0 

4 

1 

7 

3 

4 

8 

1 

2 

9 

—  10 


Ai  =  .01470  +  .00063  X  20  =  .014S82 
sin  2(pi  =  A^Ci 


log  A, 

8 

1 

7 

2 

6 

0 

—  10 

log  (7, 

0 

9 

9 

o 

8 

3 

log  sin  2(pi .  . 

9 

1 

6 

8 

4 

9 

—  10 

=  4 


8°  28'  34" 

14'  17"  first  approximation 

82  V  57 
849+     -^       —8.52.9 


A/' 
Zs  =3  2983.8 
A 


2=r  :i;>3c>.a 

„  =  .01408  +  .838  X  .00002  =  .014599 


log  Ai 
log  C, 


log  sin  202 


8 

1 

6 

4 

3 

2 

1 

0 

0 

2 

3 

5 

9 

1 

6 

6 

6 

7 

—  10 


—  10 


8°  20'  20" 

4°   13'  13"  second  approximation 
8.99  X  56 
11        '' 
2984.1 
.01408  +  .00002  X  .841  =  .014001 


<P2  = 

A,"  =  793  + 
A, 


:  838.7 


log  A3 
log  C3 


log  sin  2<t>., 


8 

1 

0 

4 

3 

8 

1 

0 

0 

2 

3 

1 

9 

1 

6 

6 

6 

9 

—  10 


—  10 


2(^3  =  8°  26'  28" 
(p3^4°   13'  14"  third  approximation 


R  =  10,000  yards  X  —  30,000  feet, 
loglog  /  =  log  Y  —  5.01765 
Y  =  A"C  tan<^ 


losr  A" I 

loi  o, 

log  tan  4>i  ■  •  ■ 
log  consta-nt. 

loglog  fi 


log  /i  . 
log  C, . 

log  C,. 
log  X .. 

log  Z,, 


0 
5 

6  I  9 

1      7 


0  j  6 
9  I  5 


0      2      3 

7  I  7  I  1 

7   !  4  i  7 


—  10 

—  10 


—  10 


Z,  =  2983.8 


log  A.," 

log  Co 

log  tan  0o . .  . 

2 
1 

8 

9 
0 

8 

2 
0 
6 

3 
2 

8 

6 
3 
0 

1 
5 
0 

—  10 

log  constant. 

5 

0 

1 

7 

6 

5 

—  10 

loglog /s 

7 

8 

1 

1 

6 

1 

—  10 

log/= 

log  Ci 

0 
0 

0 
9 

0 
9 

6 
5 

4 

8 

8 
3 

log  C3 

1 

0 

0 

2 

3 

1 

log  A- 

4 

4 

7 

7 

1 

2 

logZ3 

3 

4 

7 

4 

8 

1 

Z3 


2984.1 


From  tlie  above    work,  we    have    for  our 
final  values: 

<p  =  4°   13'  14" 
Z  =  2984.1 
log  C  —  1.00231 


Form  No.  2. 
For  Computation  of 

Column  2.    Angle  of  Departure  (if  C  be  already  correctly  known,  and  work  on  Form  No.  1 

is  therefore  unnecessary). 
Column  3.     Angle  of  Fall.  Column  4.     Time  of  flight.  Column  5.     Striking  velocity. 

Column  6.     Drift.  Column  8.     Maximum  ordinate.  Column  9.     Penetration. 


JJAiA. 

R  =  10,000  yards        X  =  30,000  feet       log  C  =  1.00231         Z  =  2984.1 
A  =  .01408  +  .00002  X  -841  =  .014()01       log  B'  =  .1000  +  .0037  X  -841  =  .1 
M  =2048  — 26  X  -841  =  2026.1     7"=  1.192  +  .049  X  -841  =  1.2332      D' — 

B  =  .0178  +  .0009  X  .841  =  01856 
,„      -no,8.f9xS6        „„.  -        Ak=.00099  +  .0004x  .841  =  .001024 


Z  =  2984.1 
7  X  .841  =  .10371 
.2332      /)'  =  24  +  2  X  -841  =  25.9 


^''=.703  +  ^:^^=838.8 


2.     Angle  of  departure,  0  =  4°   13'   14" 
sin  20  =  AC 


loc  C 

1 

8 

9 

0 
1 

1 

0 
6 

6 

2 

4 

6 

3 
3 

6 

1 
8 

9 

lo"- A 

—  10 

20... 

log  sin 

—  10 

20  =  8°  20'  28" 
3.     Angle  of  fall,  w  =  5°  21'  11" 
tan  w  =  5'  tan  0 


logB' 

log  tan  0.  . .  . 

0 

8 

8 

1 

8 

9 

0 
6 

7 

3 

8 

1 

7 
0 

7 

1 
3 

4 

—  10 

log  tan  w.  . .  . 

—  10 

4.     Time  of  flight,  T=  12.43  seconds 

T  —  CT'  sec  0 


log  C 

logT'... 
log  sec  0 

lo2  T.... 


1 

1  ! 

0 

0 

2 

3 

1 

0 

0 

9 

1 

0 

3 

0 

0 

0 

1 

1 

8 

1 

0 

9 

4 

5 

2 

5.     Striking  velocity,  Vw  =  2029  f .  s. 
tu)  =  u  cos  0  sec  w 


log  " 

log  cos  0.  .  . 
log  sec  w.  . . 


log    Via 


3 

3 

0 

6 

6 

6 

9 

9 

9 

8 

8 

2 

0 

0 

0 

1 

9 

0 

3 

3 

0 

7 

3 

8 

10 


G.     Drift,  D  =  26.5  yards. 

D  =  constant  X  : —  '  where  constant  =  — r 

COS'  0  nil 


los  constant. 

log  C= 

log  D' 

log  sec'  0  . . . 


log  1.5 

log  D  . 


7 

8 

3 

1 

4 

9 

2 

0 

0 

4 

6 

2 

1 

4 

0 

8 

2 

4 

0 

0 

0 

3 

5 

4 

1 

2 

4' 

7 

8 

9 

0 

1 

7 

6 

0 

9 

1 

4 

2 

3 

9 

8 

—  10 


As  the  value  of   — ;-  is    constant    for    the 
nh 

same  gun  for  all  ranges,  its  value  is  com- 
puted first  and  then  carried  on  as  a  constant 
through  all  the  drift  computations  for  the 


range    table    for    the     particular     gun    in 
question. 

0  =  4°   13'  14" 
log  tan  0  =  8.86803  —  10 
log  cos  0  =  9.99882—  10     log  sec  0  =  0.00118 

a;  =  5°  21'   11" 
log  tan  w  =  8.97 174  —  10     log  sec  w  =  0.00190 

8.  Maximum  ordinate,  r  =  622  feet 

r  =  A"(7  tan  0 

log  A"  . . 
log  C  . .  . 
log  tan  0 8       8      6      8      0      3      —10 

log  Y  . . . 

9.  Penetration  of  armor,  £'z=  15.59" 

E°-'  :=  '■r~-77r^  =  constant  X  v,  where 


9 

9 

2 

3 

6 

6 

1 

0 

0 

2 

3 

1 

8 

8 

6 

8 

0 

3 

■7 

1 

7 

9 

4 

0 

0 

constant  = 


w"-'' 


Kd" 


log  constant. 

7 

6 

5 

0 

9 

3 

—  10 

log  r 

3 
0 

3 
9 

0 
5 

7 
8 

3 
3 

8 
1 

log  £»•' ' 

log  E 

1 

3 

6 

9 

0 

1 

colog  1.0 

9 

8 

2 

3 

9 

1 

—  10 

log  i; 

1 

1 

9 

2 

9 

2 

1.5  is  De  Marre's  coefTicient  for  face-hard- 
ened armor  as  compared  to  simple  oil  tem- 
pered and  annealed  armor. 


As  the  value  of 


Kd"- 


is  constant  for  the 


same  gun  for  all  ranges,  its  value  is  com- 
puted first  and  then  carried  through  all  the 
penetration  computations  for  the  entire 
range  table  for  the  particular  gun  in  ques- 
tion. 

Note — In  forms  Nos.  3  and  4,  whenever  it 
becomes  necessary  to  use  the  logarithms  of 
T,V!a ,  etc.,  take  the  exact  values  of  those 
logarithms  correct  to  five  decimal  places 
from  the  work  on  this  sheet,  and  do  not  use 
the  approximate  logarithms  taken  from  the 
log  tables  for  the  values  of  the  elements 
given  here  and  correct  only  to  two  places  as 
required  for  the  range  table. 


176 


EXTERIOE  BALLISTICS 


FoBM  No.  3. 

For  Computation  of 

Column     7.  Danger  space  for  target  20  feet  high. 

Column  10.  Change  in  range  for  variation  of  ±  50  f.  s.  in  initial  velocity. 

Column  11.  Change  in  range  for  variation  of  ±  10  pounds  in  weight  of  projectile. 

Column  12.  Change  in  range  for  variation  of  ±  10  per  cent  in  density  of  air. 


7.    Danger  space,  S20  =  72  yards 
h 


S  =  icot 


cot  I 


1  + 


0 

8 

2 

3 

9 

1 

log  cot  u.  ..  . 

1 

0 

2 

8 

2 

6 

log  (-cot  a;) 

1 

8 

5 

2 

1 

7 

colog  B 

6 

0 

0 

0 

0 

0 

—  10 

h        cot  w 
3   ^     R 

=  .0071  log 

7  1 

8 

5 

2 

1 

7 

—  10 

1  +  . 0071  =  1.0071 


log  (A  cot  w) 
log  1.0071  ..  . 


log  .S'20 


1 

8 

5 

2 

1 

7 

0 

0 

0 

3 

0 

7 

1 

8 

5 

5 

2 

4 

10.     A725oK=  276  yards 

^vR 


AR^oV- 


2B 


log  Av  . 
log  ft  . . 
colog  2. 
colog  B. 


log  AR.^oV 


7 

0 

1 

0 

3 

0 

4 

0 

0 

0 

0 

0 

9 

6 

9 

8 

9 

7 

1 

7 

3 

1 

4 

2 

2 

4 

4 

0 

6 

9 

12.     A/?,oC  =  213  vards 


—  10 

—  10 


AR,r,c-- 


B 


lOB 


R 


B  —  A  =  .003959 
10B  =  .1856 


log  (B  —  A). 

log  ft 

colog  lOB...  . 

log  AftioC  . .  . 


7 

5 

9 

7 

5 

9 

4 

0 

0 

0 

0 

0 

0 

7 

3 

1 

4 

2 

2 

3 

2 

9 

0 

1 

—  10 


11.     Aft  to  for  ±  10  pounds  in  w, 
ARw  =  ±  42  yards 


SV  =  0.36^V 


For  change   in  ini- 


Aft'  =  Aft.„rx '^-j    t'^1  ^'^1-ity- 


b.    Aft"=:Afti„cX 


10 


c.    Aftw^Aft'  +  Aft" 


For  change  in 
weight. 


log  0.36.  .      . 

9 

5 

.5 

6 

3 

0 

—  10 

log  Aio 

1 

0 

0 

0 

0 

0 

colog  IC 

7 

0 

6 

0 

4 

8 

—  10 

logy 

3 

4 

6 

2 

4 

0 

log   51' 

1 

0 

7 

9 

1 

8 

5y=12  f.s.,  Ai'=. 001024 


log  Aft,or 

log  12 

colog  50. . 


log  Aft' 1 


2 

4 

4 

0 

7 

0 

1 

0 

7 

9 

1 

8 

S 

3 

0 

1 

0 

3 

1 

8 

2 

0 

9 

1 

—  10 


Aft' =  66.21 


log  AftioC 

log  10 

colog  87 


log  Aft". 


2 

3 

2 

9 

0 

1 

1 

0 

0 

0 

0 

0 

8 

0 

6 

0 

4 

8 

1 

3 

8 

9 

4 

9 

—  10 


Aft": 


=  24.52 
Aft'  =  66.21 
Aft"  =  24.52 


Aftiow  =  41.62 
Determination  of  n,  n  ^ 


y=  sin  2(p 


log  V-..  . 
log  sin  2<) 
colog  ST.., 
colog  X  . 


logn. 


6 

9 

2 

4 

8 

0 

9 

1 

6 

6 

6 

9 

8 

4 

9 

2 

1 

4 

.5 

5 

2 

2 

8 

8 

0 

1 

0 

6 

5 

1 

—  10 

—  10 

—  10 


n  —  1.278 
•2n  —  2.556 
2m— 1  =  1.556   log  (271— 1)  =0.19201 
ft  =  10000        log  ft  =  4.00000 


X  =  30000 


log  X  =  4.47712 


EAXGE  TABLES;  THEIE  COMPUTATION  AND  USE 


177 


Form  No.  4. 

For  Computation  of 

Column  13.     Wind  effect  in  range. 

Column  14.     Gun  motion  in  range. 

Columns  15  and  18.     Target  motion  effect  in  range  and  deflection. 

Column  16.     Wind  effect  in  deflection. 

Column  17.     Gun  motion  effect  in  deflection. 

Column  19.     Change  in  height  of  impact  for  variation  of  ±  100  yards  in  sight  bar. 


.13.     Wind  effect  in  range, 
AKu}  =  26.7  yards. 


AXyv=^W, 


T- 


X 


X  cos  0 


2w  —  1  '"        T 
12  X  (5080 


60  X  60  X  3 


=  6.7556 


log  n 

log  X 

log  cos  0.  .  .  . 
colog  {2n — 1) 
colog  V 


log 


nX  cos  <p 
(2n— 1)  V 


;  0 

1 

0 

6 

5 

1 

4 

4 

7 

7 

1 

2 

9 

9 

9 

8 

8 

2 

9 

8 

0 

7 

9 

9 

6 

5 

3 

7 

6 

0 

0 

9 

2 

8 

0 

4 

nX  cos  0 


(2n 


1)  V 
T 


=    8.47 

=  12.43 

3.96 


log   W,2X. 

log  3.96  . 


log  ARw 


(2n  — 1)  Y 


.       nX  cos  0 

log    Gi2X 

log  ARq  .... 


—  10 

—  10 

—  10 


0 

8 

2 

9 

6 

7 

0 

5 

9 

7 

7 

0 

1 

4 

2 

7 

3 

7 

14.     Gun  motion  effect  in  range, 
ARg^=57  yards 

nZ  cos  0    ^ 


0 

9 

2 

8 

0 

4 

0 

\  ^ 

2 

9 

6 

7 

1 

7 

5 

7 

7 

1 

15  and  18.     Target  motion  effect  in  range 
and  deflection, 
Ai?T  —  Dt=  Ti2zT  —  T^„xT  =  84  yards 


log  T^^x 
logT 


log  IRt 

=^  log  Dt  ■ 


0 

8 

2 

9 

6 

7 

1 

0 

9 

4 

5 

2 

1 

9 

2 

4 

1 

9 

16.     Wind  effect  in  deflection, 
Z>  IK  =13.9  yards 

X 


Dw=W,,, 


"Fcos0 


log  X. .  .. 

colog  V  . . 
log  sec  (p. 

log 


Y  cos  <p 


4 

4 

7 

7 

1 

0 

6 

5 

3 

7 

6 

0 

0 

0 

0 

1 

1 

8 

1 

0 

1 

5 

9 

0 

= =10.37 

Y  cos  <f> 

r=  12.43 

2.06 


log  IF,,,, 
log  2.0G . . 


log  Dw 


—  10 


0 

8 

2 

9 

6 

1 
7 

0 

3 

1 

3 

8 

7 

1 

1 

4 

3 

5 

4 

17.     Gun  motion  effect  in  deflection, 
Dg— 70.1  yards 
X 


Dg. 


Y  cos  (p 


f^iiz 


log  G... 
log 


Y  cos(f>" 


log  Dg 1 


0 

8 

2 

9 

6 

7 

1 

0 

1 

5 

9 

0 

1 

8 

4 

5 

5 

7 

19.     Change  in  height  of  impact  for  varia- 
tion of  ±  100  yards  in  sight  bar,  H  =  28  feet 
H  =  AX  tan  w        aX  =  300  feet 


log  AX 

log  tan  cj. . . 


Ion  H, 


2 

4 

7 

1 

2 

8 

9 

7 

1 

7 

4 

1 

4 

4 

8 

8 

6 

—  10 


12 


178 


EXTEEIOPt  BALLISTICS 


EXAMPLES. 

1.  For  examples  in  determining  the  angle  of  departure  corresponding  to  any- 
given  range,  the  data  in  the  range  tables  may  be  used,  computing  for  standard  atmos- 
pheric condition,  and  proceeding  to  determine  the  true  value  of  the  ballistic  coefficient 
by  successive  approximations.     (See  also  Examples  in  Chapter  8.) 

2.  As  the  process  of  successive  approximations  is  somewhat  long  for  section 
room  work,  the  following  are  given.  Given  the  data  contained  in  the  following  table, 
compute  the  corresponding  values  of  cf>,  w,  T  and  fa,  of  the  drift,  of  the  maximum 
ordinate,  and  of  the  penetration  of  armor  by  capped  projectiles  (Harveyized  armor, 
by  Davis's  formula  for  guns  A,  B,  D,  E,  G,  I,  J,  L  and  0 ;  face-hardened  armor,  by 
De  Marre's  formula  for  guns  C,  F,  H,  K,  M,  N,  P,  Q  and  R.  De  Marre's  coefficient 
=  1.5).     Atmosphere  standard. 


DATA. 

ANSWERS. 

Penetra- 

Projectile. 

Multi- 

Maxi- 

tion. 

s 

Value 
of/. 

Y. 

f.s. 

Range. 
Yds. 

plier 

for 

drift. 

0. 

03. 

T. 
Sees. 

f.s. 

Drift. 

Yds. 

mum 
ordi- 
nate. 
Ft. 

o 
u 

d.    w. 
In.  Lbs. 

1 
3'     13 

c. 

Harv. 
In. 

F.H. 
In. 

A... 

1.00 

1.004411.50 

2500 

1.0 

6° 

53.1' 

8°  26' 

7.89 

837 

7.5 

253 

0.96 

B... 

3     13 

1.00 

1.0034  2700 

3600 

1.0 

2 

48.9 

5    10 

6.61 

1094 

5.9 

181 

1.30 

C... 

4     33 

0.67 

1. 0011:2900 

3000 

1.0 

1 

15.3 

1    35 

3.71 

2043 

1.6 

55 

"iig 

D... 

5      oO 

1.00 

1.0025  3150 

4000 

1.5 

1 

53.4 

3    05 

5.57 

1511 

5.8 

126     3.40 

.... 

E... 

5      50 

0.61 

1.0024  3150 

4500 

1.5 

1 

45.7 

2    26 

5.49 

1932 

5.4 

122     4.6 

F... 

6    105 

0.61 

1.0605  2600 

14,000 

1.5 

13 

27.3 

24    17 

28.88 

1070 

183.0 

3513      .... 

'2.'9 

G... 

6    105 

1.00 

1.0022  2800 

3800 

1.0 

1 

52.6 

2    35 

5.21 

1729 

3.2 

109     5.7 

.... 

H.. 

6    105 

0.61 

1.0015  2800 

3500 

1.5 

1 

28.6 

1    46 

4.29 

2153 

3.0 

74      ....   1  7.9 

I... 

71  165 

1.00 

1.0095;2700 

7000 

1.0 

4 

45.6 

7    59 

11.78 

1243 

17.6 

566     4.6 

.... 

J... 

7'  165 

0.61 

1.0083  2700 

7500 

1.5 

4 

04.1 

5    42 

10.821631 

20.9 

473     6.4 

K.. 

8   260 

0.61 

1.00S5  2750 

8000 

1.5 

3 

59.8 

5    21 

10.961771 

21.5 

484 

"s.'i 

L... 

10  510 

1.00 

1.014112700 

9000 

1.0 

5 

31.6 

8    33 

14.14  14C1 

24.6 

811     8.6 

.... 

M.. 

10   510 

0.61 

1.0137j2700 

10000 

1.5 

5 

10.5 

6    55 

13.95[1744 

34.5 

785  '   .... 

10.4 

N.. 

12   870 

0.61 

1.11302900 

24000 

1.5 

15 

07.7 

25    01 

39.511359 

309.9 

6358      .... 

8.8 

0... 

131130 

1.00 

1.0337  2000 

10500 

1.0 

11 

32.4 

16    40 

21.901157 

59.9 

1955     9.4 

P... 

131130 

0.74  1.0316'2000 

11000 

1.5 

10 

52.1 

14    37 

21. 3611279 

82.9 

1845 

.... 

"s.'9 

Q... 

141400 

0.70i  1.057 12000 

14000 

1.5 

14 

37.1 

20    02 

28.281246 

148.4 

3246 

9.3 

R... 

14 1400 

0.70  1. 0342^2600 

14500 

1.5 

8 

41.7 

12    13 

22.10J1560 

89.3 

1975  1   

1 
1 

12.8 

3.  Given  the  data  and  results  contained  in  Example  2,  compute  the  correspond- 
ing values  of : 

1.  Danger  space  for  a  target  20  feet  high. 

2.  Change  in  range  resulting  from  a  variation  from  standard  of  ±50  f.  s.  in  the 

initial  velocity. 

3.  Change  in  range  resulting  from  a  variation  from  standard  of  ±10  per  cent 

in  the  density  of  the  atmosphere. 

4.  Change  in  range  resulting  from  a  variation  from  standard  as  given  below 

in  the  weight  of  the  projectile : 

Gun  C    ±    1  pound. 

Guns  F  and  // ±   3  pounds. 

Gun  J    ±   4  pounds. 

Gun  K ±    5  pounds. 

Guns  M,N,P,Q  i^ndiR ±10  pounds. 


EAXGE  TABLES;  THEIE  COMPUTATION  AND  USE 

ANSWERS. 


179 


Change  in  range 

Change  in  range 

Change  in  range 

Problem. 

Danger  space. 
Yds. 

for 
variation  in  V. 

for  varia- 
tion in  density. 

for 
variation  in  w. 

Yds. 

Yds. 

Yds. 

A 

45.8 
75.2 

200.6 

±  114.5 
±    74.4 
-+-    83 . 2 

^      46.3 
T    163.9 
T      63.0 

B 

C 

Hi  33.6 

D 

127.3 

±    82.3 

T    154.7 

E 

162.4 

■+-  104.4 

T    125.6 

F 

14.8 

±279.8 

H=    656.4 

-H37.9 

O 

153.5 

-4-    99 . 0 

=i=    104.7 

H 

229.5 

=h  103.2 

T      56.8 

q=43.2 

I 

47.9 

■+-  159.1 

T    284.4 

J 

67.4 

-+-  199.9 

T    214.7 

T42.2 

K 

71.8 

±211.3 

T    202.1 

^41.6 

L 

44.6 

-+-  220.4 

IP    319.9 

^I 

55.3 

±275.9 

:p    253.9 

ii=55.4 

N 

14.3 

±  522 . 0 

=F  1010.3 

T    9.2 

0 

22.3 

-+-  339 . 9 

:;:    333.7 

P 

25.6 

±389.5 

=?    290.0 

T  24 . 0 

Q 

18.3 

-H  479.8 

h:    399.0 

h:20.9 

R 

30.9 

-+-  395.4 

qi    425.2 

H=22.5 

•i.  Given  the  data  and  results  contained  in  Example  2,  compute  the  correspond- 
ing values  of  (atmospheric  conditions  being  standard)  : 


Effect  in  range  and  deflection  of  wind  components  of  12  knots. 
Effect  in  range  and  deflection  of  a  speed  of  gun  of  12  knots. 
Effect  in  range  and  deflection  of  a  speed  of  target  of  12  knots. 
Change  in  position  of  point  of  impact  in  the  vertical  plane  through  the  target 
for  a  variation  of  ±100  yards  in  the  setting  of  the  sight  in  range. 

ANSWERS. 


Value 

of  n. 

Wind. 

Speed  of  gun. 

Speed 
of  target. 

Change  of 
point  of  im- 

Problem. 

Range. 
Yds. 

Deflection. 
Yds. 

Range. 
Yds. 

Deflection. 
Yds. 

Range  and 

deflection. 

Yds. 

pact  in  ver- 
tical plane. 
Ft. 

A 

1 . 3032 
2.0577 
1.2709 
1.G929 
1 . 4028 
2.2622 
1..39S1 
1.1947 
1.7830 
1.4241 
1.3609 
1.6075 
1 . 3558 
1.8278 
1 . 5459 
1.3941 
1 . 4446 
1.4424 

17.8 
26.8 
7.8 
19.4 
14.6 

126.9 
13.8 
7.2 
43.2 
29 . 8 
27.6 
46.7 
35.0 

155.4 
70.9 
.59.0 
86.1 
63.8 

8.9 
17.6 

4.1 
11.9 

8.1 
82.9 

7.7 

3.6 
26.9 
16.7 
14.0 
27.7 
18.9 
93.2 
.3!) .  4 
30 . 8 
44.4 
35.0 

35.5 
17.8 
17.3 
18.3 
22.5 
68.1 
21.4 
21.8 
36.4 
43.3 
46 . 5 
4S.8 
59.2 

111.5 
77.0 
85.3 

105.0 
85.5 

44.4 

27.1 

21.0 

25.7 

29.0 

112.1 

27.5 

25.3 

52.7 

56 . 4 

59.1 

67.9 

7."5.4 

173.7 

10S.6 

113.5 

146 . 6 

114.3 

53.3 

44.7 

25 . 1 

37.6 

37.1 

195.1 

35.2 

29.0 

79.6 

73.1 

74.0 

95 . 5 

94.2 

266.9 

147.9 

144.0 

191.1 

149.3 

-1-    44  5 

R 

-t-    -'7   1 

C 

D 

E 

-+-      8.3 
±    16.2 

■+■    !•>  7 

F 

G 

±  135.4 

■+■    13  5 

H 

■+■      9  3 

I. ... 

-+-    4?  1 

J 

-t-    ''9  9 

K 

L 

±    28.1 
■+-    45   1 

M.. 

-J-    3(]  4 

N 

-+-  140  0 

0 

-H    8!)  8 

P 

■+-    78  2 

Q 

-+-  109  4 

R 

-^    (35  0 

CHAPTER  17. 

THE  PRACTICAL  USE  OF  THE  RANGE  TABLES, 

Range  tables.  274.  A  range  table  should  be  so  constructed  as  to  supply  all  the  data  necessary 

to  enable  the  gun  for  which  it  is  computed  to  be  properly  and  promptly  laid  in  such  a 
manner  that  its  projectiles  may  hit  a  target  whose  distance  from  the  gun  is  known. 
This  condition  is  fulfilled  by  the  official  range  table  computed  and  issued  by  the 
Bureau  of  Ordnance  for  each  of  our  naval  guns.  In  its  simplest  form,  such  a  range 
table  consists  of  a  tabular  statement  of  the  values  of  the  elements  of  a  series  of  com- 
puted trajectories  pertaining  to  successive  horizontal  ranges,  within  the  possible 
limits  of  elevation  of  the  gun  as  mounted,  which  is  generally  about  15°  for  naval 
guns,  with  such  ranges  taken  as  arguments  and  with  the  ranges  and  corresponding 
data  disposed  in  regular  order  for  ready  reference,  so  that  any  desired  range  may  be 
quickly  found  in  the  table,  and  from  it  all  the  corresponding  elements  of  the  required 
trajectory.  In  other  words,  complete  and  accurate  knowledge  of  all  the  elements  of 
the  trajectory  for  each  range  is  essential  to  the  efficient  use  of  the  gun,  and  to  this 
must  be  added  complete  data  as  to  the  effect  upon  the  range  of  any  reasonable  varia- 
tions in  such  of  the  ballistic  elements  as  are  liable  to  differ  in  service  from  those 
standard  values  for  which  the  table  is  computed.  There  must  also  be  added  the  neces- 
sary data  to  show  the  variations  in  range  and  deflection  resulting  from  the  velocity 
of  the  wind  and  from  motion  of  the  gun  and  target.  We  have  seen  in  the  preceding 
chapters  how  to  compute  all  this  data. 
Constants  275.  The  constants  upon  which  a  range  table  is  based  we  have  seen  to  be  the 

variations.  Caliber,  weight  and  coefficient  of  form  of  the  projectile,  that  is,  the  factors  from 
which  the  value  of  the  ballistic  coefficient  is  computed ;  the  initial  velocity ;  and  the 
jump,  this  last  being  habitually  considered  as  practically  non-existent  in  service 
unless  there  is  reason  to  believe  to  the  contrary  in  special  cases.  The  initial  velocity, 
as  well  as  the  characteristics  of  the  projectile,  constitute  features  of  the  original 
design  of  each  particular  type  of  gun ;  and,  although  the  values  of  some  of  them  may 
-be  somewhat  modified  as  the  result  of  preliminary  experimental  firings,  they  are  fixed 
quantities  when  the  question  of  sight  graduations  and  of  range  table  data  is  under 
consideration.  Of  course  the  initial  velocity  and  weight  of  projectile  may  vary  some- 
what from  their  assigned  standard  values,  the  amount  of  variation  from  round  to 
round  depending  upon  the  regularity  of  the  powder,  the  care  taken  in  the  manu- 
facture and  inspection  of  the  ammunition  and  in  putting  up  the  charges,  etc.  Two 
very  possible  and  important  causes  of  variation  in  the  initial  velocity  are  variations 
from  standard  in  the  temperature  of  the  powder,  and  drying  out  of  the  volatiles  from 
the  powder.  Both  of  these  causes  have  a  very  marked  effect  upon  the  initial  velocity, 
and  to  overcome  them  efforts  are  made  to  keep  the  magazines  at  a  steady  temperature 
and  all  at  the  same  temperature;  while  each  charge  is  kept  in  an  air-tight  case  in 
order  to  prevent  evaporation  of  the  solvent  remaining  in  the  powder  when  it  is  issued 
to  service.  It  may  be  pointed  out  that  it  is  more  important  that  all  magazine 
temperatures  shall  be  kept  the  same  throughout  the  ship  than  that  they  should  be  kept 
at  the  standard  temperature.  If  the  charges  for  all  guns  are  at  the  same  temperature, 
then,  so  far  as  that  point  is  concerned,  the  guns  will  all  shoot  alike  if  the  battery  has 
been  properly  calibrated;  and  the  spotter  can  readily  allow  for  variations  from 
standard ;  but  if  one  magazine  is  at  a  high  temperature  and  another  at  a  low,  then 


EAXGE  TABLES;  THEIR  COMPUTATION^  A>^D  USE 


181 


standard 
projectiles. 


the  guns  involved  will  have  different  errors  resulting  from  this  cause,  and  the  spotter 
cannot  hope  to  get  the  salvos  bunched  when  the  sights  of  the  several  guns  are  set  for 
tlie  same  range. 

276.  The  sights  are  marked  and  the  range  table  computed  for  the  mean  initial   ^^s^t 

°  .  markings. 

velocity  and  the  mean  weight  of  projectile,  and  these  are  made  identical  with  the 
fixed  standard  values  as  given  in  the  range  table.  In  preparing  projectiles  for  issue 
to  the  service  great  care  is  exercised  to  bring  the  weight  of  each  one  to  standard  so 
that  this  cause  of  variation  in  range  may  not  appear. 

277.  Up  to  within  the  past  few  years  (that  is,  up  to  the  adoption  of  the  long 
pointed  projectile)  the  value  of  the  coefficient  of  form  was  taken  as  unity.  This  was 
its  value,  for  which  the  ballistic  tables  were  computed,  for  the  type  of  projectile 
then  standard  in  service,  as  described  in  the  preceding  chapters  of  this  book.  With 
the  adoption  of  the  long  pointed  projectile,  however,  the  value  of  the  coefficient  of 
form  has  dropped  below  unity,  and  for  the  several  guns  and  projectiles  covered  by 
the  Eange  and  Ballistic  Tables  published  for  use  with  this  text  book,  its  value  ranges 
from  1.00  for  blunt  pointed  projectiles  (radius  of  ogival  of  2.5  calibers)  to  from  0.74 
to  0.61  for  long  pointed  projectiles  (radius  of  ogival  of  7  calibers).  Its  value,  what- 
ever it  is,  must  be  used  in  computing  the  value  of  the  ballistic  coefficient^,  so  long  as 
the  present  ballistic  tables  are  used.  Perhaps  it  may  be  advisable  some  day  to  recom- 
pute the  ballistic  tables  with  the  long  pointed  projectile  as  the  standard  projectile  of 
the  tables,  in  which  case  the  coefficient  of  form  of  such  a  projectile  would  then  become 
unity  for  computations  with  the  new  tables ;  and  a  coefficient  of  form  whose  value  is 
greater  than  unity  would  have  to  be  used  for  computations  involving  the  blunt-nosed 
projectiles.  Such  recomputation  of  the  tables  has  not  yet  been  made,  however,  and  it 
is  unlikely  that  it  will  be  done  unless  progress  in  the  development  of  ordnance  makes 
recomputation  necessary  by  raising  service  initial  velocities  above  the  present  upper 
limit  of  the  ballistic  tables. 

278.  To  show  the  results  obtained  by  the  adoption  of  the  long  pointed  projectile, 
let  us  compare  the  range  tables  for  the  7"  gun  of  2700  f .  s.  initial  velocity,  weight  of 
projectile  165  pounds,  for  a  range  of  7000  yards,  for  each  of  the  two  values  of  the 
coefficient  of  form.    The  two  range  tables  give : 


Value  of  c. 

1.00 
0.61 


Rangp  for  an 
angle  of  ele- 
vation of 
about  15°. 
Yards. 

13100 

16900 


Time  of 
flight. 
Sees. 

11.76 
9.89 


Striking 
velocity. 

f.  s.  ■ 

1247 
1690 


Danger  space 

for  target 

20'  high. 

Yards. 

'    48 
76 


Maximum 

ordinate. 

Feet. 

563 
39.5 


Penetration  in 

Harve\'ized 

armor. 

Incnes. 

4.6 
6.7 


From  what  has  been  studied  in  the  preceding  chapters,  a  glance  at  the  above  figures 
will  show  at  once  how  vastly  improved  the  performance  of  the  gun  has  been  in  every  • 
particular  by  the  introduction  of  the  long  pointed  projectile. 

279.  After  the  preceding  preliminary  remarks  it  is  possible  to  proceed  to  a  care- 
ful consideration  of  the  uses  to  which  the  range  tables  may  be  put  in  service,  and 
this  will  now  be  done,  column  by  column. 

280.  Explanatory  Notes. — The  explanatory  notes  at  the  beginning  of  each  range  Explanatory 
table  are  in  general  a  statement  of  the  standard  conditions  for  which  the  data  in  the  "angl  tlbies. 
columns  is  computed,  and  of  the  methods  by  which  it  is  computed.    There  is  one  item 

given  therein  which  is  used  in  practical  computations  aboard  ship,  however,  and  that 
is  the  information  in  regard  to  the  effect  upon  the  initial  velocity  of  variations  in  the 
temperature  of  the  powder.  The  note  in  every  case  gives  the  standard  temperature 
of  the  powder,  which  is  generally  taken  as  90°  F.;  and  then  states  that  a  variation 
from  this  standard  temperature  of  ±10°  causes  a  variation  in  initial  velocity  of  about 
±35  foot-seconds  in  the  initial  velocity  in  most  cases,  although  in  some  cases  the 
variation  in  initial  velocity  for  that  amount  of  variation  in  temperature  is  ±  20  foot- 


183 


EXTEEIOR  BALLISTICS 


Col.  1,  range. 


Col.  2,  *. 


To  lay  gun 

at  given 

angle  of 

elevation. 


seconds  instead  of  ±35  foot-seconds.  For  instance,  with  our  standard  problem  12" 
gun,  we  see  that  the  variation  in  initial  velocity  for  a  variation  of  ±10°  from  standard 
is  ±35  foot-seconds.  Therefore,  if  the  temperature  of  the  charge  were  80°  F.,  our 
initial  velocity  would  be  2865  foot-seconds  and  not  2900  foot-seconds.  If  the  tempera- 
ture of  the  charge  were  100°  F.,  the  initial  velocity  would  be  2935  foot-seconds.  A 
variation  of  ±5°  in  the  temperature  of  the  charge  gives  a  proportionate  change  in 

35 


the  initial  velocity,  that  is,  ±-j-rX5=:  ±17.5  foot-seconds 


and  if  the  temperature 
of  the  charge  were  77°  F.,  we  would  have  a  resultant  initial  velocity  of 

35  X  13 


2900- 


10 


=  2854.5  foot-seconds 


and  similarly  for  other  variations. 

281.  Column  1.  Range. — As  already  explained,  this  is  the  argument  column  of 
the  table.  The  data  in  the  other  columns  is  obtained  by  computation  for  ranges 
beginning  at  one  thousand  yards  and  increasing  by  five-hundred  yard  increments,  and 
that  for  intermediate  ranges  by  interpolation  from  the  computed  results  (using 
second  or  higher  differences  where  such  use  would  affect  the  results).  Therefore,  to 
obtain  the  value  of  any  element  corresponding  to  a  range  lying  between  the  tabulated 
ranges  as  given  in  Column  1,  proceed  by  interpolation  by  the  ordinary  rules  of 
proportion. 

282.  Column  2.  Angle  of  Departure  =  Angle  of  Elevation  +  Jump. — As  has 
been  said,  although  jump  must  be  watched  for  and  considered  in  any  special  case 
where  it  may  be  suspected  or  found  to  exist,  still  it  is  normally  practically  nonexistent 
in  service,  and  the  angle  of  departure  and  angle  of  elevation  coincide  for  horizontal 
trajectories.    C 

283.  To  lay  the  gun  at  any  desired  angle  of  elevation,  the  sights  being  marked 
in  yards  and  not  in  degrees;  find  the  angle  of  elevation  in  Column  2,  and  the  corre- 
sponding range  from  Column  1.  Set  the  sight  at  this  range,  point  at  the  target,  and 
the  gun  will  then  be  elevated  at  the  desired  angle.  An  example  of  this  kind  is  given 
in  paragraphs  188,  189,  190  and  191  of  Chapter  11.  As  there  seen,  this  process  is 
necessary  when  shooting  at  an  object  that  is  materially  elevated  or  depressed  relative 
to  the  horizontal  plane  through  the  gun. 

Let  us  now  see  how  correctly  the  range  tables  may  be  used  to  determine  the 
proper  angle  of  elevation  to  be  used  to  hit  an  elevated  target;  assuming  the  theory  of 
the  rigidity  of  the  trajectory  as  true  within  the  angular  limits  probable  with  naval 
gun  mounts.  For  this  purpose  we  will  consider  the  problem  solved  in  paragraph  188 
of  Chapter  11,  Avhich  was  for  the  12"  gun;  7  =  2900  f.  s.;  w  =  870  pounds;  c  =  0.61; 
horizontal  distance  =  10,000  yards;  elevation  of  the  target=1500  feet  above  the  gun  ; 
barometer  =  29.00";  thermometer  =  90°  F.  In  paragraph  188  we  computed  that  the 
correct  angle  of  elevation  for  this  case  is  4°   08.1'. 

Now  let  us  use  Column  12  and  correct  for  atmospheric  conditions,  for  which, 
from  Table  IV,  the  multiplier  is  -1-0.79 ;  from  which  we  have  that  for  a  sight  setting 
of  10,000  yards  the  shell  would  range  10000+  (215  x  .79)  =10169.85  yards.  There- 
fore in  order  to  make  the  shell  travel  10,000  yards  we  must  set  the  sight  in  range  for 
9830.15  yards.  From  the  range  table,  by  interpolation,  the  proper  angle  of  departure 
for  this  range  is  4°  07.9'.  If  we  use  this  for  the  angle  of  elevation  desired  (instead 
of  the  computed  value  of  4°  08.1')  we  will  have  an  error  of  2',  that  is,  of  about  7  yards 
short. 

Xow  if  we  solve  the  triangle  to  determine  the  actual  distance  from  the  gun  to  the 
target  in  a  straight  line,  we  will  find  it  to  be  9857.8  yards  (by  use  of  traverse  tables), 
using  the  horizontal  distance  corrected  for  atmospheric  conditions  as  the  base.    From 


EANGE  TABLES;  THEIE  CGMPUTATIOX  AND  USE  183 

the  rano:e  table,  the  angle  of  departure  for  this  range  is  4°  08.6'.  If  we  use  this  as  the 
desired  angle  of  elevation  we  will  have  an  error  of  5'  in  elevation,  or  of  about  17 
yards  over. 

The  above  processes  show,  for  this  individual  case,  the  degree  of  inaccuracy  that 
would  enter  from  the  use  of  the  range  table  for  this  pur])ose  :  and  the  errors  introduced 
above  are  not  great  enough  to  prevent  the  first  shot  from  falling  within  reasonable 
spotting  distance.  That  this  will  be  the  case  under  all  circumstances  cannot  be  pre- 
dicted, and  each  individual  case  must  be  considered  on  its  own  merits.  For  instance, 
it  is  evident  that  the  horizontal  distance  could  not  be  used  in  attacking  an  aeroplane 
at  a  high  angle  with  a  gun  so  mounted  as  to  enable  such  angles  to  be  used. 

It  is  to  be  noted  that,  when  we  use  the  range  table  as  described  above,  assuming 
that  the  trajectory  be  rigid,  we  get  the  same  results  whether  the  target  be  elevated 
above  or  depressed  below  the  horizontal  plane  of  the  gun.  This  shows  at  once  that  the 
method  is  not  theoretically  perfect. 

284.  For  subcaliber  practice  a  one-pounder  gun  is  mounted  in  oi'  on  a  turret  gun,   subcaiiber 

sis'litin&r* 

the  axes  of  the  two  guns  being  parallel;  and  it  is  desired  to  point  the  pair,  using  the 
sights  of  the  turret  gun,  so  that  the  one-pounder  projectile  will  hit  a  target  at  a  known 
range.  A  combination  range  table  for  this  purpose  is  given  on  page  34  of  the 
Gunnery  Instructions,  1913,  but  if  this  be  not  available  we  may  proceed  as  follows 
(and  this  is  the  method  by  which  the  table  referred  to  was  prepared)  for  our  standard 
problem  12"  gun :  Suppose  that  our  subcaliber  target  is  1750  yards  away.  Look  in 
the  range  table  for  the  one-pounder  gun  that  is  to  be  used,  and  it  will  be  found  that 
the  angle  of  departure  for  that  gun  for  1750  yards  is  4°  13'.  The  problem  then 
becomes  similar  to  that  stated  in  the  preceding  paragraph.  Look  in  the  12"  range 
tables  and  it  will  be  seen  that  for  that  gun  the  angle  of  departure  of  4°  13'  corre- 
sponds to  a  range  of  10,000  yards.  Set  the  12"  sights  at  10,000  yards,  point  at  the 
target  with  those  sights,  and  the  guns  will  then  be  so  elevated  that  the  one-pounder 
shell  should  hit  at  1750  yards. 

285.  A  problem  that  has  come  up  a  number  of  times  in  our  service  has  been  to   Use  of  same 

sislits  for 

use  the  si2:hts  marked  for  the  initial  velocitv  due  to  a  full  charge  when  firing  with    varying 

'  .  .    .  ,  r        .  initial 

reduced  charges  and  correspondingly  reduced  initial  velocities.  Suppose,  for  in-  velocities, 
stance,  we  have  our  standard  problem  12"  gun  and  are  to  fire  it  with  the  reduced 
charge  which  gives  2100  f.  s.  initial  velocity  instead  of  the  regular  2900  f.  s.  given 
by  the  full  charge.  Proceeding  by  the  methods  already  explained,  we  would  compute 
the  angles  of  departure  necessary  to  give  the  desired  ranges  with  the  reduced  velocity. 
Having  tabulated  the  results,  suppose  we  find  that,  for  a  given  range  A,  the  proper 
angle  of  departure  is  6°  08'  42".  Looking  in  the  range  tallies  for  2900  f.  s.,  we  find 
that  the  proper  range  at  which  to  set  the  sights  in  order  to  get  this  angle  of  departure 
is  13,300  yards,  and  if  we  set  the  full  charge  sights  for  that  range  we  should  hit  at  A 
yards  with  the  reduced  charge.  The  results  should  be  tabulated  for  all  probable 
ranges,  and  the  resulting  table  used  by  the  spotter ;  or  else  paper  sight  scales  may  be 
prepared  for  the  reduced  velocity  and  pasted  on  the  sights  over  the  old  scales. 

286.  Knowing  the  possible  amAe  of  elevation  resulting  from  the  mechanical    Roii  as  umit- 

•-  >-  '^  '^  ing  iire. 

features  of  the  mount,  Column  2  will  also  show  what  degree  of  roll  will  necessarily 
throw  the  line  of  sight  off  the  target  at,  say,  the  bottom  of  the  roll.  We  have  seen 
that,  with  our  standard  problem  13"  gun  at  10,000  yards,  we  must  have  an  angle  of 
departure  of  4°  13'.  Most  turret  guns  cannot  be  elevated  more  than  15°,  therefore 
when  a  ship  has  rolled  so  as  to  depress  her  guns  10°  the  total  angle  of  elevation 
required  of  the  mount  a:t  the  bottom  of  the  roll  would  be  10° -f  4°  13'  =  14°  13', 
which  is  so  near  the  limit  of  elevation  of  15°  as  to  show  that  it  wouLd  probably  be 
impossible  to  fire  on  the  bottom  of  the  roll  under  these  conditions. 


184  EXTERIOR  BALLISTICS 

Col.  3,  «  287.  Column  3.    Angle  of  Fall. — This  column  will  give  information  as  to  the 

angle  of  inclination  of  the  axis  of  the  projectile  to  the  face  of  the  target  at  any  given 
range  which  is  known  as  the  "  angle  of  impact " ;  and,  also,  its  inclination  to  the  sur- 
face of  the  water  at  the  point  of  fall,  and  hence  of  the  probability  of  a  ricochet. 

288.  It  is  also  used  for  computing  the  value  of  the  danger  space  by  the  use  of 
the  formulge  given  in  Chapter  5. 
Col.  4,r  289.  Column  4.    Time  of  Flight. — This  column  gives  one  of  the  elements  that 

enter  into  the  time  interval  necessary  between  successive  shots  from  the  same  gun, 
that  is,  between  salvos.    For  instance,  with  our  standard  problem  12"  gun  at  10,000 
't    u    ^        yards,  we  see  that  the  time  of  flight  is  12.43  seconds.     The  time  that  must  elapse 
(  ,  between  salvos  is  therefore  13  seconds  plus  the  additional  time  it  takes  to  spot  the  shot 

'  and  get  ready  for  the  next  one.    It  will  be  seen  from  this  that  the  longer  the  range  the 

longer  must  be  the  interval  between  salvos,  provided  they  be  properly  spotted.  At 
the  longer  ranges  ordinarily  used  the  time  between  two  successive  salvos  as  deter- 
mined above  is  ample  for  loading,  so  that  the  loading  interval  does  not  enter  under 
those  conditions  in  determining  the  time  between  salvos.  At  short  ranges,  however, 
the  loading  interval  might  be  greater  than  the  time  as  determined  above,  and  in  such 
cases  the  loading  interval  would  necessarily  determine  the  salvo  interval. 

290.  The  information  contained  in  this  column  is  also  necessary  to  determine 
the  setting  for  time  fuses  Avhen  using  shrapnel.  In  order  to  have  this  information 
readily  available  at  the  gun,  the  range  scales  of  the  sights  generally  bear  correspond- 
ing time  scales  showing  the  time  of  flight  in  seconds  corresponding  to  any  given  range. 

291.  When  guns  of  different  calibers  are  being  fired  together  at  the  same  target^ 
or  when  different  ships  are  firing  together  at  the  same  target,  the  information  con- 
tained in  this  column  aids  the  spotter  to  identify  the  splashes  of  the  shot  which  he  is 
spotting.  An  assistant  should  start  a  stop  watch  at  the  instant  the  guns  in  question 
are  fired,  and  then  the  shot  that  splash  at  the  end  of  the  tabulated  time  of  flight  are  in 
all  probability  the  ones  in  which  the  spotter  is  interesed. 

292.  Also,  suppose  a  ranging  shot  goes  well  over,  and  its  time  of  flight  is 
measured.  Its  excess  over  the  time  of  flight  for  the  given  range  is  a  check  on  the 
amount  of  spot  necessary  to  bring  the  next  shot  down  to  the  target,  and  a  check  like 
this  does  much  to  increase  the  confidence  of  the  spotter  in  ordering  a  large  change  in 
sight  setting  under  such  conditions. 

Col.  5,  vu  ^^^-  Column  5.    Striking  Velocity. — The  only  practical  use  of  this  column  is  to 

enable  a  judgment  to  be  formed  of  the  effect  of  the  projectile  at  any  given  range,  and 
hence  as  to  the  advisability  of  using  a  given  gun  for  a  given  purpose  of  attack  at  a 
given  range. 
Col.  6,  drift.  294.  Column    6.      Drift. — The    drift   being   accurately   compensated   in    sight 

setting  with  all  modern  telescopic  sights,  at  all  ranges,  the  data  in  this  column  is  of 
no  special  value  with  modern  guns,  but  of  general  interest  only.  With  the  old  bar 
sights,  however,  in  which  the  drift  compensation  was  made  by  inclining  the  sight  bar 
at  a  permanent  angle,  the  compensation  for  drift  was  correct  at  a  single  range  only, 
and  at  all  other  ranges  the  uncompensated  portion  of  the  drift  had  to  be  allowed  for 
by  the  use  of  the  sliding  leaf.  In  such  a  case,  if  the  accurate  compensation  be  at 
7000  yards,  we  would  have  to  know  the  amount  of  drift  compensated  by  the  perma- 
nent angle  at  any  other  range  at  which  we  wish  to  shoot,  say  10,000  yards,  and  the 
difference  between  that  and  the  tabulated  value  of  the  drift  at  10,000  yards  would 
have  to  be  compensated  by  the  use  of  the  sliding  leaf. 
Col.  7,  danger  295.  Column  7.    Danger  Space  for  a  Target  20  Feet  High. — This  column  is  very 

frequently  used.  In  the  first  place,  it  gives  us  a  quick  general  idea  of  the  probability 
of  making  a  hit,  for  the  greater  the  danger  space  the  greater  the  chance  of  success. 


space. 


RANGE  TABLES;  THEIB  COMPUTATION  AND  USE  185 

It  also  shows  us  the  "  danger  range  "  or  "  point  blank  range  " ;  that  is,  the  maximum 
range  for  which  the  projectile  never  rises  higher  than  the  top  of  the  target.  (Com- 
pare with  the  same  information  as  given  in  Column  8.)  For  our  standard  problem 
12"  gun  we  see  that  this  is  the  case  up  to  and  including  2100  yards,  at  which  range 
the  maximum  ordinate  is  20  feet.  For  targets  of  other  heights  than  20  feet,  the 
danger  space  may  be  determined  within  reasonable  limits  of  height  by  a  simple 
proportion  from  the  data  contained  in  this  column.  Thus,  for  our  standard  problem 
12"  gun,  at  10,000  yards,  the  danger  space  for  a  target  30  feet  high  would  be  about 

30 
72  X  -^77=  108  yards.    If  the  height  be  so  great  that  these  results  are  not  sufficiently 

accurate,  then  the  new  danger  space  must  be  computed  by  the  formulfe  given,  the 
shorter  one,  S  =  h  cot  w,  being  ordinarily  sufficiently 'accurate. 

A  wrong  conception  of  the  danger  space  is  often  acquired,  namely,  that  it  may  be 
defined  as  the  distance  from  the  target  to  the  point  of  fall  of  a  projectile  that  just 
touches  the  top.  of  the  target.  This  conception  may  apply  fairly  well  at  long  ranges, 
but  a  reference  to  Column  7  of  the  range  table  for  short  ranges  will  show  that  it  does 
not  fit  the  data  given  in  that  column.  Considering  Column  19  of  the  range  table, 
which  gives  the  change  in  the  height  of  the  point  of  impact  in  the  vertical  plane 
through  the  target  resulting  from  a  variation  of  100  yards  in  the  setting  of  the  sight 
in  range,  for  long  ranges  the  height  of  the  target  divided  by  the  figures  given  in 
Column  19  gives  the  same  result  as  the  danger  space  given  in  Column  7,  but  this  is  not 
the  case  for  short  ranges.  Bearing  in  mind  that  the  danger  space  is  the  distance  that 
the  target  can  be  moved  from  the  point  of  fall  directly  toward  the  gun  at  the  given 
range,  and  still  be  hit,  it  will  be  seen  that,  as  the  range  is  reduced,  as  soon  as  the 
maximum  ordinate  becomes  equal  to  the  height  of  the  target,  then  the  target  may  be 
moved  all  the  way  to  the  gun  and  still  have  the  trajectory  pierce  the  screen,  so  that 
the  danger  space  is  then  equal  to  the  range,  and  at  the  point  where  this  happens  we 
have  the  danger  range.  As  a  matter  of  fact  Column  19  should  be  used  for  all  practical 
computations,  but  there  should  be  no  confusion  of  thought  as  to  the  relation  existing 
between  Columns  7  and  19. 

296.  At  long  ranges  a  knowledge  of  the  danger  space  shows  immediately  how  far 
beyond  the  target  a  shot  will  fall  that  just  touches  the  top  of  the  target  screen. 

297.  Column  8.     Maximum  Ordinate. — This  column  is  valuable  for  use  in   coi.  8,  y 
determining  the  value  of  the  altitude  factor,  /,  in  making  ballistic  computations  to 
obtain  approximate  solutions. 

298.  Column  9.    Penetration  of  Armor  (Harveyized  or  Face  Hardened)  with   coi.  9.  Ei 
Capped  Projectiles. — The  data  in  this  column  is  of  value  only  in  determining  the 
probable  efficiency  of  attack  upon  armor  at  different  ranges.    The  figures  given  in  the 
column  are  for  normal  impact,  and  the  angle  of  fall  (as  given  in  Column  3)  must  be 

taken  into  account  in  considering  this  subject  in  order  to  determine  the  angle  of  im- 
pact, the  "  angle  of  impact "  against  any  surface  being  that  angle  less  than  90°  between 
the  axis  of  the  projectile  at  the  moment  of  impact  and  the  surface  in  question. 
Thus,  for  a  vertical  armor  plate  at  the  same  level  as  the  gun,  the  angle  of  impact  is 
the  complement  of  the  angle  of  fall.  For  elevated  or  depressed  targets  the  angle  of 
inclination  at  the  striking  point  must  be  used  instead  of  the  angle  of  fall  given  in  the 
table.  Up  to  a  certain  critical  angle  we  get  penetration  in  the  usual  manner,  although 
a  pronounced  increase  in  the  angle  of  impact  probably  reduces  the  efficiency  of  pene- 
tration. The  critical  angle  referred  to  is  that  at  which  the  ix)int  of  the  projectile 
ceases  to  bite,  and  we  no  longer  have  the  penetrative  effect  due  to  the  shape  of  the 
point  but  simply  the  smashing  effect  due  to  the  momentum,  which  effect  is  of  course 


186  EXTEEIOR  BALLISTICS 

small  as  compared  to  penetration  proper.    This  subject  is  not  particularly  well  under- 
stood up  to  the  present  time. 

299.  The  tables  which  give  the  penetration  in  Harveyized  armor  were  computed 
*     before  the  present  form  of  face-hardened  armor  came  into  general  use.     A  rough 

approximation  to  the  penetration  of  face-hardened  armor  may  be  obtained  from  those 
tables  by  multiplying  the  penetration  in  Harveyized  armor  by  0.8. 

300.  For  our  standard  problem  12"  gun,  at  10,000  yards  range,  the  angle  of  fall, 
as  given  by  the  range  table,  is  5°  21'.  The  angle  of  impact  against  a  vertical  side 
armor  plate  would  therefore  be  84°  39'.  The  angle  of  impact  against  a  protective 
deck  plate  inclined  to  the  horizontal  at  an  angle  of  15°  would  be  20°  21',  which  is 
probably  very  near  or  less  than  the  biting  angle,  and  little  if  any  penetrative  effect 
could  be  expected ;  in  other  words,  the  protective  deck  would  probably  deflect  the  pro- 
jectile, thus  fulfilling  the  purpose  for  which  it  was  designed.  If,  however,  the  ships 
being  on  parallel  courses,  the  target  ship  were  rolled  10°  towards  the  gun  at  the 
moment  of  impact,  the  angle  of  impact  against  the  vertical  side  armor  would  l^e 
74°  39';  and  that  against  the  protective  deck  plate  would  be  30°  21', 

301.  For  an  elevated  target,  a  vertical  armor  plate,  taking  the  problem  given  in 
paragraph  188  of  Chapter  11,  the  angle  of  inclination  to  the  horizontal  at  the  target 
is  ^=2°  20.1',  and  the  angle  of  impact  would  therefore  be  87°  39.9'.  For  the  prob- 
lem given  in  paragraph  189  of  Chapter  11,  the  angle  of  impact  against  the  vertical 
plate  would  similarly  be  82°   03',  as  we  have  ^=  (  —  )  7°   57'  in  this  case. 

Col.  10.  302.  Colunm  10.     Change  of  Range  for  a  Variation  of    ±  50  Foot-Seconds 

'■^*"  Initial  Velocity. — A  number  of  causes  tend  to  produce  variations  in  initial  velocity, 
one  being  a  variation  in  the  temperature  of  the  charge,  which  has  already  been  dis- 
cussed. The  volatiles  in  the  powder  may  dry  out,  giving  a  resultant  quicker  burning 
powder,  with  an  increase  in  both  pressure  and  initial  velocity.  A  damp  powder  will 
burn  more  slowly  and  give  a  reduced  initial  velocity.  Slight  deterioration  in  the 
powder  not  sufficient  in  amount  or  of  a  character  to  cause  danger  may  reduce  the 
initial  velocity  (any  deterioration  that  causes  an  increase  in  the  initial  velocity  will 
cause  increased  pressure,  and  should  be  looked  upon  as  dangerous).  There  is  no 
means  of  determining  the  amount  of  variation  of  initial  velocity  due  to  these  last  two 
causes  except  experimental  firing.  Firing  at  a  given  range  under  known  conditions, 
with  all  known  causes  of  variation  eliminated,  as  will  hereafter  be  explained  in  the 
discussion  of  calibration  practice,  and  a  comparison  of  the  resulting  actual  range  with 
that  given  in  the  range  tables  for  the  given  angle  of  elevation,  would  give  an  approxi- 
mate idea  of  any  such  change  in  initial  velocity,  by  working  backwards  in  this  column. 
303.  For  our  standard  problem  12"  gun,  suppose  we  know  that  our  charge  was  at 
a  temperature  of  100°  F. ;  that  the  solvent  had  dried  out  enough  to  cause  an  increase 
in  initial  velocity  of  15  f.  s.,  and  that  deterioration  of  the  non-dangerous  kind  had 
reduced  the  initial  velocity  by  20  f .  s.    Our  final  initial  velocity  would  then  be : 

Standard  initial  velocity 2900  f.  s. 

Variation  due  to  temperature  of  charge.  ...  -f- 35  f .  s. 
Variation  due  to  drying  out  of  volatiles.  ...  -|- 15  f .  s. 
Variation  due  to  deterioration  of  powder.  . .  —  20  f.  s. 

Total   variation    4-30  f.  s..  .      30  f .  s. 

Actual  initial  velocity 2930  f.  s. 

And  at  10,000  yards  range,  this  variation  in  initial  velocity  would  give  us,  from 

277 
Colunm  10,  a  resulting  actual  range  of  10000  +  —..-^  X  30  =  10166  yards,  or  the  gun 

OO 


EANGE  TABLES;  THEIR  COMPUTATION  A:NrD  USE  187 

would  shoot  166  yards  over  the  target  unless  allowance  was  made  for  this  variation 
by  setting  the  sight  at  9834  yards. 

304.  These  figures  show  the  importance  of  keeping  all  powder  charges  at  a  con- 
stant temperature,  and  all  the  charges  in  the  ship,  particularly  for  guns  of  the  same 
caliber,  at  the  same  temperature ;  and  also  for  keeping  the  volatiles  from  drying  out 
and  for  keeping  the  powder  from  becoming  damp ;  as  well  as  for  using  care  to  prevent 
deterioration,  even  if  not  of  the  dangerous  kind.  They  also  show  how  seriously 
results  may  be  affected  by  keeping  a  powder  charge  in  a  hot  gun  before  being  fired 
long  enough  to  let  the  temperature  of  the  gun  materially  raise  that  of  the  charge; 
it  being  particularly  necessary  to  look  out  for  this  point  when  carrying  on  a  calibra- 
tion practice. 

305.  Working  back  with  the  problem  given  above,  suppose  the  gun  had  been 
fired  at  an  angle  of  elevation  of  4°  13'  14",  that  is,  sighted  for  10,000  yards,  and  that 
the  point  of  fall  had  been  accurately  determined  by  triangulation ;  and,  that,  after  the 
observed  results  had  been  reduced  to  standard  conditions,  the  data  showed  an  actual 
range  of  10,100  yards,  or  100  yards  in  excess  of  normal.  This  would  tend  to  show 
that  a  drying  out  of  volatiles  had  taken  place  sufficient  to  give  an  increase  in  initial 

velocity  of  — ir;^^:—  =18  f.  s.    If  this  work  be  deemed  reliable,  we  could  then  figure 

277 

on  an  initial  velocity  of  2918  f.  s.  for  further  firing.  . 

306.  Column  11.    Change  of  Range  for  a  Variation  of  ±Aw  Pounds  in  Weight  coi.  ii, 
of  Projectile. — All  projectiles,  before  issue  to  service,  should  be  brought  to  standard 
weight,  and  it  will  be  found  that  this  has  usually  been  done,  and  that  there  is  ordi- 
narily little  use  for  the  data  contained  in  this  column.     However,  if  for  calibration 

or  for  any  other  form  of  experimental  firing,  we  find  that  the  projectiles  are  not  of 
standard  weight,  and  that  it  is  not  practicable  or  convenient  to  make  them  so,  we  can 
reduce  the  results  to  standard  by  the  use  of  this  column.  (The  regular  service  pro- 
jectile for  the  14"  gun  is  subject  to  a  tolerance  of  ±4  pounds  in  weight;  that  is,  these 
projectiles  may  weigh  anywhere  from  1396  to  1404  pounds.  For  ordinary  firing 
this  small  variation  in  weight  is  considered  as  immaterial.) 

307.  It  is  important  to  note  that,  where  there  is  no  sign  prefixed  to  the  entry  in 
this  column,  an  increase  in  the  weight  of  the  projectile  causes  a  decrease  in  range 
and  the  reverse ;  but  if  the  entry  in  the  column  carries  a  negative  sign  (as  is  the  case 
in  some  parts  of  the  table  for  the  6"  gun  for  which  F  =  2600,  w  =  105,  and  c  =  0.61), 
then  an  increase  in  weight  causes  an  increase  in  range  at  all  ranges  for  which  the 
negative  sign  appears  in  this  column. 

308.  With  our  standard  problem  12"  gun  at  10,000  yards,  a  projectile  weighing 

42  V  7 
877  pounds  would  travel  10000-     V:     =9970  yards. 

309.  With  the  6"  gun,  for  which  7  =  2600  f.  s.,  w  =  105  jDounds,  and  c  =  0.61, 
if  the  shell  weighed  110  pounds,  for  a  set  range  of  12,400  yards,  the  travel  would 

actually  be  12400 +  ^4^^  =12430  yards. 

310.  The  physical  reason  why,  under  some  conditions,  an  increase  in  weight  of 
projectile  gives  a  decrease  in  range  at  one  range  and  an  increase  at  another  may  be 
readily  understood  if  we  remember  that  the  effect  upon  the  range  of  an  increase  in  the 
weight  of  the  projectile  is  the  result  of  two  entirely  independent  causes.  The  first 
acts  entirely  before  the  projectile  leaves  the  gun,  and  an  increase  of  weight  thus 
acting  always  causes  a  decrease  in  the  initial  velocity,  and  hence,  so  far  as  this  part 
of  the  effect  alone  is  concerned,  an  increase  in  weight  would  always  cause  a  decrease 
in  range.    The  second  part  of  the  effect,  however,  which  acts  entirely  outside  the  gun. 


188  EXTEEIOE  BALLISTICS 

is  that  due  to  the  momentum  stored  in  the  moving  projectile;  that  is,  it  depends  upon 
the  weight.  As  the  weight  is  a  factor  in  Mayevski's  expression  for  retardation,  we  see 
that  an  increase  in  weight  increases  the  power  of  the  projectile  to  overcome  the 
atmospheric  resistance,  and  hence  increases  the  range.  Therefore  of  two  similar  pro- 
jectiles, differing  somewhat  in  weight  but  leaving  the  gun  with  the  same  initial 
velocity,  the  heavier  would  travel  the  further;  and,  so  far  as  this  part  of  the  effect 
alone  is  concerned,  an  increase  in  weight  of  the  projectile  would  always  give  an 
increase  in  the  range.  We  do  not  have  equal  initial  velocities  in  the  case  under 
consideration,  however,  and  the  increase  in  the  weight  of  the  projectile  acts  first  to 
decrease  the  initial  velocity,  and  then,  after  leaving  the  gun,  to  make  the  projectile 
travel  further  than  would  one  of  standard  weight  if  fired  with  the  same  reduced 
initial  velocity.  It  can  readily  be  conceived  from  this  line  of  reasoning  that,  under 
some  conditions,  this  second  effect  might  more  than  balance  that  due  to  loss  of 
initial  velocity,  and  in  all  such  cases  an  increase  in  the  weight  of  the  shell  will  increase 
the  range.  The  proper  sign  for  the  data  in  Column  11  is  determined  by  a  careful 
consideration  of  the  relative  values  of  the  two  parts  in  the  formula  from  which  the 
data  is  derived. 

311.  An  inspection  of  the  range  table  for  the  6"  gun  referred  to  in  paragraph 
309  above  will  show  that  for  ranges  from  1000  yards  to  10,800  yards  an  increase  in 
the  weight  of  the  projectile  will  cause  a  decrease  in  the  range ;  at  10,900  yards  it  will 
cause  no  change  in  the  range;  and  from  11,000  yards  up  an  increase  in  range  will 
result. 

312.  For  the  standard  problem  12"  gun,  through  the  entire  table,  from  1000 
to  24,000  yards,  it  will  be  seen  that  increase  in  weight  of  projectile  causes  decrease 
in  range.  It  is  of  interest  to  note,  however,  that  from  1000  yards  to  about  12,000 
yards  this  decrease  in  range  increases  with  the  range  from  a  minimum  of  8  yards  at 
1000  yards  to  a  maximum  of  42  yards  at  about  12,000  yards;  and  that  from  about 
12,000  yards  up  the  variation  decreases  until,  at  the  highest  point  of  the  table,  24,000 
yards,  the  decrease  in  range,  due  to  an  overweight  of  10  pounds  in  the  projectile,  is 
only  12  yards.  Apparently  there  is  some  theoretical  point  beyond  the  upper  limit  of 
the  range  table  at  which  this  quantity  would  change  sign,  and  beyond  which  an  in- 
crease of  10  pounds  in  the  weight  of  the  projectile  would  cause  an  increase  in  range, 
in  a  manner  similar  to  that  discussed  above  in  regard  to  the  6"  gun.  This  point  is  of 
course  of  no  practical  value  in  connection  with  the  12"  gun,  whereas  it  must  neces- 
sarily be  taken  into  account  in  dealing  with  the  6"  gun. 

coi.^12,  313.  Column  12.    Change  of  Range  for  a  Variation  of  Density  of  Air  of  ±10 

Per  Cent. — As  has  been  stated,  the  range  tables  have  been  computed  for  a  standard 
atmosphere  of  half -saturated  air,  for  59°  F.  (15°  C.)  and  29.53"  (750  mm.)  baro- 
metric height.  Of  course  this  exact  atmospheric  condition  will  rarely  exist  in  actual 
firing,  and  the  data  in  this  column  has  been  computed  to  enable  allowance  to  be  made 
for  variations  from  standard  density.  It  is  of  course  easier  for  a  projectile  to  travel 
through  a  less  dense  than  through  a  more  dense  medium ;  and  if  the  air  be  below  the 
standard  density  the  range  will  therefore  be  greater  than  the  standard  range,  etc. 

314.  For  our  standard  problem  12"  gun,  range  10,000  yards,  suppose  the  barom- 
eter stood  at  31.00"  and  the  thermometer  at  50°  F.  From  Table  III  of  the  Ballistic 
Tables,  for  those  conditions,  the  value  of  8  is  1.069,  or  the  air  is  6.9  per  cent  above 

215  X  6  9 
the  standard  density,  and  our  actual  range  would  be  10000—  — — — ^  =9582  yards. 

If  the  barometer  stood  at  29.00"  and  the  thermometer  at  96°  F.,  the  value  of  8  would 
be  0.910,  that  is,  the  air  would  be  9  per  cent  below  standard  density,  and  the  actual 

range  would  be  10000+  ^1|^  =10194  yards. 


A^ioc 


EANGE  TABLES;  THEIK  COMPUTATIOX  AND  USE  189 

315.  Or  for  the  same  problem,  using  Table  IV  of  the  Ballistic  Tables,  in  the 
first  case  the  actual  range  would  be  10000  —  215x0.69  =  9852  yards,  and  in  the 
second  case  it  would  be  10000  +  215x0.9  =  10194  yards;  which  process  is  shorter 
but  cannot  be  understood  unless  the  first  and  longer  method  has  first  been  com- 
prehended. 

316.  This  column  is  designated  as  referring  to  variations  in  the  density  of  the 

air,  this  factor  (8)  being  the  one  going  to  make  up  the  value  of  the  ballistic  coefficient 

that  is  most  apt  to  vary.     If  we  remember  that  the  formula  for  the  value  of  the 

fw 
ballistic  coefficient  is  C=  J  -j^  ?  we  can  see  that  a  variation  of  any  given  per  cent  in 

any  one  of  the  factors  gives  the  same  numerical  percentage  change  in  the  value  of  C, 
remembering  that  an  increase  in  /  or  w  gives  an  increase  in  C,  and  an  increase  in  8, 
c  or  d^  gives  a  decrease  in  C.  Changes  in  w  cannot  be  handled  in  this  way,  owing  to 
the  resultant  change  in  initial  velocity  already  explained.  Changes  in  /  are  not  often 
known  in  such  shape  as  to  make  it  convenient  to  handle  them  by  this  method,  that  is, 
by  the  use  of  the  data  given  in  this  column,  although  it  could  of  course  be  done  in  this 
way  were  the  percentage  variation  in  the  value  of  /  known.  Also  c  and  d  are  ordi- 
narily constant,  thus  leaving  8  as  the  only  one  of  the  factors  of  the  ballistic  coefficient 
that  would  ordinarily  be  considered  from  this  point  of  view.  From  the  range  table 
for  this  gun  computed  for  c=1.00,  for  3000  yards  range,  we  have  an  entry  of  84 
yards  in  Column  12.    If  now  the  value  of  c  becomes  0.95,  we  have  a  decrease  of  5  per 

84  X  5 

cent  in  the  value  of  c,  and  therefore  our  range  would  increase  to  3000  H r-^r —  =3042 

yards.     Xow,  similarly,  if  we  use  the  table  for  the  same  gun,  but  with  c  =  O.Gl,  we 

have  in  Column  12  of  that  table,  for  a  range  of  3000  yards,  the  data  GO  yards.    Xow 

5 
if  the  value  of  c  increases  from  0.61  to  0.66,  we  have  an  increase  of -j— =.082,  or 

8.2  per  cent;  and  the  decreased  range  resulting  from  this  change  would  be 

3000-  ^i^A^  =2945.88. 

Theoretically  we  should  be  able,  starting  at  a  given  range  in  the  table  for  c=1.00,  to 
reduce  the  range  by  the  correction  from  Column  12  for  a  variation  of  39  per  cent, 
and  thus  get  the  range  for  a  projectile  for  which  c  =  0.61  that  would  correspond  to 
the  range  of  3000  yards  for  the  projectile  for  which  c  =  1.00.  Then  starting  with 
this  new  range  in  the  table  for  which  c=0.61,  and  applying  the  correction  from 

39 

Column  12  for  a  variation  of  ^  =.64,  or  64  per  cent,  we  should  get  the  original 

range  from  which  we  started  as  the  corresponding  range  for  the  projectile  for  which 
c  =  1.00.  This  will  not  work  out  very  closely,  however,  because  the  percentage  change 
in  such  a  case  is  too  large  to  be  handled  by  the  use  of  data  such  as  that  contained  in 
Column  12,  which  is  computed  by  the  use  of  a  formula  based  on  differential  incre- 
ments. 39  per  cent  and  64  per  cent  manifestly  cannot  be  considered  as  such  incre- 
ments. 


190 


EXTEEIOE  BALLISTICS 


Col.  13,  wind 

in  range. 


317.  Column  13.  Change  of  Range  for  Wind  Component  in  Plane  of  Fire  of 
12  Knots. — This  column  is  constantly  used.  For  our  standard  proolem  12"  gun,  at 
10,000  yards,  a  wind  blowing  directly  from  the  target  to  the  gun  with  a  velocity  of  12 
knots  would  decrease  the  actual  range  27  yards,  and  would  increase  it  the  same 


A^A//^ 


/27' 


^i^ 


f 


/^'V 


l^/nd 


Figure  23. 


amount  if  blowing  the  other  way.  Suppose  the  line  of  fire  were  37°  true,  and  the 
wind  were  blowing  from  210°  true  with  a  velocity  of  25  knots.  Then  the  wind  com- 
ponent in  the  line  of  fire  would  be  25  cos  53°,  or  (by  use  of  the  traverse  tables)  15 


knots,  and  the  range  would  be  increased 


27x15 
12 


=  3-1  yards  by  this  component. 


EANGE  TABLES;  THEIR  COMPUTATION  AND  USE 


191 


318    Column  14.    Chansre  of  Range  for  Motion  of  Gun  in  Plane  of  Fire  of  12   coi.  i4,  gun 

■,■,-,,,  motion  in 

Knots. — Tliis  column  is  also  constantly  used.    For  our  standard  problem  13    gun,  at   range. 
10,000  yards,  if  the  gun  be  moving  at  12  knots  directly  towards  the  target,  it  will 
overshoot  57  yards  unless  the  motion  of  the  gun  be  allowed  for  in  pointing;  and  if 


V 


< 


/  ^      45" 


> 


Figure  34. 


moving  in  tlie  opposite  direction  it  would  undershoot  by  the  same  amount.  If  the 
line  of  fire  be  3?°  true,  and  the  firing  ship  be  steaming  315°  true  at  20  knots,  the 
component  speed  in  the  line  of  fire  would  be  20  cos  82°,  or  (by  the  use  of  the  traverse 

tables)  2.8  knots  towards  the  target,  and  the  gun  would  overshoot  ■ — ^  '  ■  =13.3 
yards. 


192 


EXTEEIOR  BALLISTICS 


Col.  15,  tar- 
get motion 
in  range. 


319.  Column  15.  Change  of  Range  for  Motion  of  Target  in  Plane  of  Fire  of 
12  Knots. — This  column  is  also  constantly  used.  For  our  standard  problem  12"  gun 
at  10,000  yards,  if  the  target  be  steaming  directly  towards  the  gun  at  12  knots,  the 
gun  would  overshoot  the  mark  84  yards  unless  the  motion  were  allowed  for  in  point- 
ing; and  if  it  were  steaming  at  the  same  rate  in  the  opposite  direction  it  would  under- 


MorfK 


Figure  25. 


shoot  by  the  same  amount.  If  the  line  of  fire  were  37°  true,  and  the  target  were 
steaming  175°  true  at  23  knots,  the  component  of  motion  in  the  line  of  fire  would  be 
23  cos  42°,  or  (by  the  use  of  the  traverse  tables)  17.1  knots  toward  the  gun,  and  the 


gun  would  overshoot 


84x17.1 
12 


=  143.6  yards. 


RAXGE  TABLES:  THEIE  COMPUTATION"  AND  USE 


193 


320.  Column  16.  Deviation  for  Lateral  Wind  Component  of  12  Knots. — This 
column  is  also  constantly  used.  For  our  standard  problem  12"  gun  at  10,000  yards, 
if  the  wind  were  blowing  perpendicular  to  the  line  of  fire  and  across  it  from  right  to 
left,  with  a  velocity  of  12  knots,  the  shot  would  fall  14  yards  to  the  left  of  the  target 


Col.  16,  wind 
in  deflection. 


A&rfA. 


Figure  26. 


unless  the  effect  of  the  wind  were  allowed  for  in  pointing.  If  the  line  of  fire  be  37" 
true,  and  the  wind  be  blowing  from  310°  true  at  23  knots,  the  win^  component  per- 
pendicular to  the  line  of  fire  would  be  23  sin  87°,  or  (by  the  use  of  the  traverse  tables) 

23  knots,  and  the  shot  would  fall  — — —  =27  yards  to  the  right  of  the  target. 


13 


194 


EXTERIOE  BALLISTICS 


Col.  17,  gun 
motion  in 
deflection. 


321.  Column  17.  Deviation  for  Lateral  Motion  of  Gun  Perpendicular  to  Line 
of  Fire,  Speed  12  Knots. — This  column  is  also  constantly  used.  For  our  standard 
problem  12"  gun,  at  10,000  yards,  if  the  gun  be  moving  at  13  knots  perpendicular  to 
the  line  of  fire,  and  from  right  to  left,  the  shot  would  fall  70  yards  to  the  left  of  the 


Figure  27. 


target  unless  the  motion  were  allowed  for  in  pointing.  If  the  line  of  fire  be  37°  true, 
and  the  firing  ship  be  steaming  100°  true  at  21  knots,  the  component  of  this  motion 
perpendicular  to  the  line  of  fire  would  be  21  sin  63°,  or  (by  use  of  the  traverse  tables) 

18.7x70 


18.7  knots  to  the  right,  and  the  shot  would  fall 


12 


109  yards  to  the  right. 


EANGE  TABLES;  THEIE  COMPUTATION  AND  USE 


195 


322.  Column  18.  Deviation  for  Lateral  Motion  of  Target  Perpendicular  to 
Line  of  Fire,  Speed  12  Knots. — This  eohimn  is  also  constantly  used.  Note  that  the 
change  of  range  in  yards  for  the  given  speed  when  the  target  is  moving  in  the  line  of 
fire  is  always  the  same  numerically  as  the  deviation  in  yards  for  the  same  speed  M'hen 
the  motion  is  perpendicular  to  the  line  of  fire.  This  is  manifestly  correct,  as  the 
motion  of  the  target,  unlike  any  of  the  other  motions  considered,  has  no  effect  upon 
the  actual  motion  of  the  projectile  relative  to  the  ground.  This  motion  of  the  target 
simply  removes  the  target  from  the  point  aimed  at  by  an  amount  equal  to  the  dis- 


/Y^r/K 


Col.  18,  tar- 
get motion  in 
deflection. 


Figure  28. 


tance  traveled  by  it  during  the  time  of  flight.  For  our  standard  problem  12"  gun  at 
10,000  yards,  if  the  target  be  moving  at  12  knots  perpendicular  to  the  line  of  fire, 
from  right  to  left,  the  shot  would  fall  84  yards  astern  of,  that  is,  to  the  right  of  the 
target  unless  allowance  were  made  for  this  motion  in  pointing.  If  the  line  of  fire  be 
37°  true,  and  the  target  be  steaming  180°  true  at  20  knots,  then  the  component  of 
motion  perpendicular  to  the  line  of  fire  would  be  20  sin  37°,  or  (by  the  use  of  the 


traverse  tables),  12  knots  to  the  left,  and  the  shot  would  fall 
the  left. 


84x12 
12 


=  84  yards  to 


196 


EXTERIOR  BALLISTICS 


Relation  be- 
tween deflec- 
tion in  yards 
and  in  knots. 


323.  There  is  another  most  important  use  to  which  the  data  contained  in  this 
column  is  constantly  put,  and  that  is  the  determination  of  the  i^oint  at  which  to  set 
the  deflection  scale  of  the  sight  to  compensate  for  any  known  deviation  in  yards. 
Deflection  scales  could  just  as  properly  he  marked  in  any  units,  say  parts  of  an  inch 
motion  of  the  sliding  leaf  either  way  from  the  central  position,  or  simply  in  arhitrary 
divisions  of  convenient  size ;  and  some  such  method  was  formerly  employed  before  the 
present  more  scientific  and  accurate  methods  of  pointing  were  introduced.  What- 
ever the  system  of  marking  the  deflection  scales,  the  essential  point  is  that  there  must 
be  some  simple  and  convenient  means  of  determining  quickly  how  many  divisions 
change  in  the  set  of  the  deflection  scale  is  necessary  to  correct  a  deviation  of  a  known 
number  of  yards  at  any  given  range.  It  has  therefore  been  found  most  convenient  to 
mark  the  deflection  scale  in  "  knots,"  meaning  "  knots  speed  of  target."  and  to  make 
the  size  of  the  divisions  such  that  setting  the  scale  over  by  12  knots,  that  is,  by  12  of 
the  divisions,  will  produce  at  any  given  range  the  number  of  yards  deviation  shown  in 
Column  18  for  that  range.  Our  telescopic  sights  have  their  deflection  scales  marked 
in  this  way.  To  avoid  the  confusion  that  was  found  to  arise  from  the  necessity  for 
using  the  words  "  right "  and  "  left "  in  giving  orders  for  sight  setting,  the  mark 
of  zero  deflection  for  the  sight  is  now  commonly  marked  as  "  50  knots,"  and  to  shift 
the  point  of  fall  of  the  shot  to  the  left  we  lower  the  reading  of  the  deflection  scale 
("  left"  and  "  lower"  both  begin  with  the  letter  /),  and  to  shift  the  point  of  fall  of 
the  shot  to  the  right  we  raise  the  reading  of  the  deflection  scale  ("right"  and 
"  raise  "  both  begin  with  the  letter  r) .  For  our  standard  problem  12"  gun,  at  10,000 
yards,  if  we  wish  to  correct  a  deflection  of  84  yards  left,  we  wish  to  shift  the  point  of 
fall  of  the  shot  that  distance  to  the  right,  and  we  accordingly  set  the  deflection  scale 
at  62  knots.  To  correct  a  deflection  of  81  yards  right,  we  would  similarly  set  the  scale 
at  38  knots.  If  the  deflection  had  been  25  yards,  we  would  have  set  the  deflection 
25x12 


scale  over 


81 


:3.6  knots,  say  4  knots;  and  if  we  were  correcting  an  error  to  the 


right  (the  original  deflection  setting  having  been  50),  we  would  set  the  sight  on  46 
knots  on  the  deflection  scale;  whereas  had  the  original  error  been  to  the  left  the 
setting  would  be  54  knots.* 

324.  Column  19.  Change  in  Height  of  Impact  for  Variation  of  ±  100  Yards 
"of  pofnt'of  in  Sight  Bar. — This  column  is  also  frequently  used.  For  our  standard  problem  12" 
gun,  at  10,000  yards,  suppose  the  shot  wer-e  striking  at  an  estimated  distance  of  50 
feet  above  the  target,  and  we  want  to  know  how  much  to  change  the  setting  of  the 


Col.   19,  ver- 
tical position 


Manifestly,  we  would  lower  the  sight  in  range  by 


50x100 

28 


=  172 


sight  in  range  to  hit. 
yards. 

The  above  is  of  value  in  shooting  at  objects  on  shore,  where  it  is  in  some  cases 
easier  to  estimate  the  vertical  distance  of  the  point  of  impact  from  the  target  than  it 
is  the  error  in  range ;  and  also  in  direct  flight  spotting  where  the  shot  can  be  seen  to 
pass  over  the  screen. 

325.  We  also  see  that,  by  the  use  of  Column  19,  a  change,  with  our  standard 

problem  12"  gun  of  — ^^ —  =72  yards  in  range  will  change  the  vertical  position  of 

^8  ^ 

the  point  of  impact  20  feet,  at  10,000  yards  range,  that  is,  from  the  top  to  the  bottom, 
or  vice  versa,  of  a  target  screen  20  feet  high,  which  is  the  danger  space  at  that  range 
for  such  a  target,  and  corresponds  with  the  danger  space  as  given  in  Column  7. 


*  See  Appendix  C  for  a  description  of  the  arbitrary  deflection  scale  for  sights,  which 
has  recently  been  adopted  for  service  use. 


EANGE  TABLES;  THEIR  COMPUTATION  AND  USE 


197 


326.  If  we  were  shooting  at  10,000  yards  on  the  sight  bar,  with  the  same  gun, 

and  gave  a  spot  of  •"  up  200,"  this  would  raise  the  position  of  the  point  of  impact  on 

28  X  200 
the  target  screen,  or  rather  in  the  vertical  plane  through  it,  a  distance  of  — — —  =56 

feet. 

327.  We  are  now  in  a  position  to  proceed  to  the  solution  of  some  every-day   Real  wind 
practical  problems  by  the  use  of  the  range  tables,  and  for  the  first  one  we  will  take  a   p^obfim! 
ship  steaming  southwest  at  18  knots,  which  wishes  to  fire  a  12"  gun  (7  =  2900  f.  s., 

w  =  S70  pounds,  c  =  0.61)  at  another  ship  that  is  8000  vards  distant  and  bears  30°  off 
the  port  bow  of  the  firing  ship  at  the  moment  of  firing.    The  target  ship  is  steaming 


Figure  29, 

west  at  22  knots,  and  the  real  wind  is  blowing  from  the  south  at  20  knots.  The 
barometer  is  at  29.67"  and  the  thermometer  at  20°  F.  The  temperature  of  the 
powder  is  70°  F.  Drying  out  of  volatiles  has  raised  the  initial  velocity  25  f.  s.,  and 
dampness  of  powder  has  reduced  it  10  f.  s.  The  shell  weighs  875  pounds.  How 
must  the  sights  be  set  to  hit?  * 

For  given  atmospheric  conditions  8  =  1.089,  that  is,  the  air  is  8.9  per  cent  over 
standard  density. 

Use  traverse  tables  for  all  resolutions  of  speeds. 

35 
Powder  is  20°  below  standard,  reducing  V  by  ^rr  X  20 —  70  f .  s. 

'  *         -    lU 

Drying  out  of  volatiles  increases  U  by +  25  f .  s. 

Dampness  of  powder  reduces  U  by — 10  f.  s. 

Total  variation  of  initial  velocity  from  standard —55  f .  s. 

*  See  Appendix  B  for  a  description  of  the  Farnsworth  Gun  Error  Computer,  by  the  use 
of  which  these  problems  may  be  solved  mechanically. 


198 


EXTEEIOR  BALLISTICS 


Cause  of  variation 

Affects. 

Formulae. 

Range. 

Deflection. 

Speed  of  or  varia- 
tion in — 

Short. 
Yds. 

Over. 

Yds. 

Right. 
Yds. 

Left. 
Yds. 

Gun - 

Target - 

Wind - 

Range 

Deflection.. 

Range 

Deflection.. 

Range 

Deflection.. 

Range 

Range 

Range 

18  cos  30  X  ^  =  15.6  X  ^ 

18  sin  30X^  =  9Xy| 

22  cos  75  X  ^  =  5.7  X  -^ 

22sin75x^  =  21.3X-§ 

20  cos  15X^=10.3X  -^ 

20  sin  15  X  32"  =  5.2  x  ~ 
^r.       229 

39 
S.9xf 

30.9 

27.3 

251.9 

19.5 

121.0 

61.1 

42.0 
3.5 

115.4 

Initial  velocity  . .  . 
IV 

s 

4.50.6 
61.1 

61.1 

45.5 

115.4 
45.5 

Point  of  fall  of  shot  if  uncorrec 

ted 

389.5 
yards 
short. 

69.9 

yards 
left. 

Heal  wind 

and  speed 

problem. 


To  correct  a  deflection  of  69.9  yards,  set  deflection  scale  to  right — '--^ —  =12.9 

65 
knots  of  scale.    Therefore  to  point  correctly,  set  sights  at 

In  range   8389.5  yards 

In  deflection 62.9  knots 

or,  to  nearest  graduations  of  sight  scales,  remembering  to  shoot  short  rather  than  over, 

In  range    8350  yards 

In  deflection    63  knots. 

328.  A  12"  gun  (7  =  2900  f.  s.,  w  =  870  pounds,  c  =  0.61)  mounted  on  board  a 
ship  steaming  45°  (magnetic)  at  18  knots,  is  to  be  fired  at  a  target  ship  on  the  star- 
board bow  of  the  firing  ship  and  steaming  315°  (magnetic)  at  14  knots,  at  the 
moment  when  the  firing  ship  is  9530  yards  from  the  point  of  intersection  of  the  two 
courses,  and  the  target  ship  is  5500  yards  from  the  same  point.  The  barometer  is  at 
28.25"  and  the  thermometer  at  80°  F.  The  temperature  of  the  powder  is  105°  F. 
Dampness  of  the  powder  has  reduced  the  initial  velocity,  at  standard  temperature,  to 
2875  f.  s.  The  shell  is  7.5  pounds  over  weight.  There  is  a  real  wind  blowing  from 
260°  by  compass  (Dev.  10°  E.)  at  20  knots.    How  should  the  sights  be  set  to  hit? 


EANGE  TABLES;  THEIR  COMPUTATION  AND  USE  199 


By  use  of  the  traverse  tables,  at  the  moment  of  firing,  the  target  will  be  30°  on 
the  starboard  bow  of  the  firing  ship,  distant  11,000  yards. 


//arr^ 


vs^-  ^rL<y 


/^rr 


/       \ 


\*=^ 


/ 


3/S-* 


N*^ 


;l«^ 


Figure  30. 


15 
Temperature  of  powder  35  x  Y7^ +  52.5  f .  s. 

Dampness    —25.0  f.  s. 

Total  variation  in  initial  velocity +27.5  f.  s. 

For  the  given  atmospheric  conditions,  8  =  0.916,  and  the  air  is  therefore  8.4  per 
cent  below  standard  density. 

Use  the  traverse  tables  for  all  resolutions  of  speeds. 


200 


EXTEEIOE  BALLISTICS 


Cause  of  error. 

Affects. 

Formulae. 

Range. 

Deflection. 

Variation  in  or 
speed  of — 

Short. 
Yds. 

Over. 
Yds. 

Right. 
Yds. 

Left. 

Yds. 

Gun - 

Range ..... 
Deflection.. 
Range ..... 
Deflection.. 

Range 

Deflection. . 

Range 

Range 

Range 

18  cos  30  X  -^  =  93  cos  30 

18  sin  30  X  ^=115.5  sin  30 

94 
14  cos  60  X  -p7  =  109.7  cos  60 

94 
14  sin  60Xj^  —  109.7  sin  60 

32 
20  cos  15  X  ^  —  53.3  cos  15 

20  sin  15  X  ^  =  28.3  sin  15 

31.5 

80.5 

54.8 

51.5 

163.4 

215.0 

95.0 
7.3 

Target - 

Wind - 

57.8 

I 
Initial  velocity.. .  . 

w 

5 

31.5 

565.2 
31.5 

102.3 

57.8 

57.8 

Point  of  fall  of  shnt  if  imcorref 

ted 

.533.7 
yards 
over. 

44.5 
yards 
right. 

To  correct  for  a  deflection  of  44.5  yards,  set  deflection  scale  to  left  by 

1  9 

-^  X  44.5  =  5.7  knots 
94 

Therefore  to  point  correctly,  set  the  sights 

In  range  for 10466.3  yards 

In  deflection  for 44.3  knots 

or,  to  nearest  graduations  of  scales,  remembering  to  shoot  short  rather  than  over, 

In  range  for 10450  yards 

In  deflection    44  knots 

329.  In  all  the  preceding  discussions  relative  to  the  wind,  both  in  Chapter  14 
and  in  this  chapter,  we  have  dealt  with  the  real  wind,  and  it  is  now  time  to  take  up  the 
discussion  of  the  apparent  wind.  The  difference  between  the  two  must  always  be 
clearly  borne  in  mind.  The  real  ivind  is  the  wind  that  is  actually  blowing ;  that  is, 
as  it  would  be  recorded  by  a  stationary  observer;  while  the  apparent  ivind  is  the  wind 
that  appears  to  be  blowing  to  an  observer  on  board  a  moving  ship.  Thus,  in  Figure 
31,  let  W  be  the  velocity  of  the  real  wind,  bloAving  at  an  angle  a°  with  the  direction 
of  motion  of  the  ship,  and  let  G  be  the  motion  of  the  ship  (in  the  same  units  as  TF), 


RANGE  TABLES;  THEIR  COMPUTATION  AND  USE 


201 


Then,  if  the  figure  be  drawn  to  scale,  W  will  represent  the  velocity  of  the  apparent 
ivind,  and  a'°  its  direction.  Or  the  solution  may  be  made  by  the  ordinary  rules  of 
plane  trigonometry. 


Figure  31. 

330.  Columns  13  and  16  of  the  range  tables  are  computed  for  real  wind,  and 
equations  (211)  and  (212)  were  therefore  used  for  the  purpose,  and  these  columns 
were  therefore  primarily  computed  for  use  in  correcting  for  the  effects  of  a  known  real 
wind.  The  same  columns  may  be  used,  under  some  circumstances,  for  correcting  for 
the  effects  of  an  apparent  wind,  as  will  now  be  explained. 

331.  Equations  (213)  and  (21-1)  give  the  total  effect  upon  the  range  and 
deviation  of  Gx  and  Gz,  respectively,  but  instead  of  using  them  (that  is.  Columns  14 
and  17  of  the  range  table),  we  may  proceed  in  another  way.  The  horizontal  velocity 
of  the  ship,  which  is  added  to  the  projectile's  proper  velocity,  would  add  to  the  range 
the  distance  GxT,  in  which  T  is  the  time  of  flight,  if  it  were  not  for  the  retardation 
caused  by  the  resistance  of  the  air.  But  the  reduction  of  this  added  distance  by  air 
resistance  is  exactly  equal  to  the  change  of  range  that  would  be  caused  by  a  wind 
component  of  Gx.  The  reasoning  is  similar  for  deflection.  Consequently  if,  in 
determining  the  wind  effect,  we  take  account  of  the  direction  and  velocity  of  the  wind 
relative  to  the  moving  ship,  that  is,  of  the  apparent  wind  instead  of  the  real  wind, 
in  so  doing  we  are  including  part  of  the  effect  of  the  ship's  motion,  and  the  remaining 
effect  of  that  motion  must  be  found,  not  by  (213)  and  (214),  but  by 

AX  0  =  GxT  (220) 

Do=G,T  (221) 

Observe  that  the  change  of  range  due  to  the  apparent  wind  which  results  from  the 
motion  of  the  ship,  Gx,  is  by  (211) 

2n-l  V 


Gx   T- 


so  that,  if  we  take  account  of  this  apparent  wind,  we  must  use  GxT  to  correct  the 
range  for  the  motion  Gx  in  order  that  the  sum  of  the  two  corrections  may  be  the  true 
effect  of  the  motion  given  by  (213)  ;  and  similarly  for  the  lateral  motion. 

332.  The  change  in  range  and  the  lateral  deviation  due  to  wind,  as  given  in   Real  and  ap- 
Columns  13  and  16  of  the  range  tables  are  those  for  an  actual  or  real  wind,  and  the 

values  in  those  colunms  are  computed  for  the  condition  that  both  gun  and  target  are 
stationary  in  the  water.  Column  13  shows  the  number  of  yards  a  shot  would  have  its 
range,  as  given  in  Column  1,  increased  or  decreased  by  a  real  wind  of  12  knots  an  hour 
blowing  directly  with  or  directly  against  its  flight.  Column  16  shows  the  number  of 
yards  the  shell  would  be  driven  to  the  right  or  left,  with  the  wind,  by  a  real  wind  of 
12  knots  an  hour  blowing  perpendicular  to  the  line  of  fire  during  the  time  the  shell  is 
in  the  air  traveling  from  the  gun  to  the  point  of  fall,  that  is,  during  the  time  of  flight 
as  given  in  Column  4,  for  the  corresponding  range  as  given  in  Column  1. 

333.  If  our  standard  problem  12"  gun  be  fired  abeam,  at  a  stationary  target 
10,000  yards  away,  on  a  calm  day,  while  the  ship  is  steaming  at  12  knots  an  hour,  the 
time  of  flight  would  be   (from  Column  4)   12.43  seconds,  but  the  shell  would  not 


202  EXTERIOR  BALLISTICS 

advance  during  its  flight  as  far  in  the  direction  of  the  course  of  the  ship  as  would  the 
ship  herself,  because  the  initial  sideways  motion  of  the  shell  due  to  the  motion  of  the 
gun  in  that  direction  would  be  retarded  by  the  resistance  of  the  air  to  such  sideways 
motion  of  the  shell  after  leaving  the  gun.  In  its  sideways  motion  the  shell  has  to 
overcome  this  air  resistance.  For  example,  for  the  above  gun  and  conditions,  the 
ship,  during  the  time  of  ilight,  would  travel  84  yards  perpendicular  to  the  line  of 
fire  (Column  18),  but  a  wind  effect  equal  to  the  speed  of  the  ship,  but  in  the  opposite 
direction,  would  reduce  the  sideways  motion  of  the  shell  in  space  by  14  yards 
(Column  16) .  Therefore  the  sideways  motion  of  the  shell  in  space  due  to  the  speed  of 
the  ship  would  be  84  —  14  =  70  yards,  which  is  the  figure  given  in  Column  17. 

334.  Again,  if  both  ship  and  target  were  stationary,  the  other  conditions  being 
as  given  above,  except  that  a  real  wind  of  13  knots  is  blowing  directly  across  the  line  of 
fire;  we  would  then  see,  from  Column  16,  that  the  shell  would  be  blown  sideways  dur- 
ing flight,  or  deflected,  by  14  yards  in  the  direction  in  which  the  wind  is  blowing,  and 
this  is  the  same  amount  as  the  difference  between  the  travel  of  the  ship  and  the  travel 
of  the  shell  in  the  direction  of  the  course  as  given  in  the  preceding  paragraph.  It 
will  thus  be  seen  that  Column  17  in  the  range  table  allows  for  that  portion  of  the 
apparent  wind  which  is  produced  by  the  speed  of  the  ship  through  still  air.  Hence 
to  use  Columns  13  and  16  for  an  apparent  wind,  which  is  the  algebraic  sum  of  the 
speed  of  the  ship  and  of  the  velocity  of  the  real  wind,  and  Columns  14  and  17  for  the 
motion  of  the  gun,  would  be  to  correct  twice  for  that  portion  of  the  apparent  wind 
which  is  produced  by  the  ship  steaming  in  still  air.  The  practical  method  of  using 
the  tables  for  apparent  wind  is  further  discussed  in  paragraph  338  of  this  chapter. 

335.  As  an  example  of  the  above,  an  inspection  of  the  range  table  for  our 
standard  problem  12"  gun  for  10,000  yards  shows  the  following  data : 

Error  in  Yards. 

(a)  Gun  fired  in  vacuum  as  far  as  resistance  of  air  is  concerned,  sliip 

steaming  at  12  knots  towards  or  away  from  target  (Col.  15)  . .   84  Over  or  Short 

(b)  Gun  fired  in  vacuum  as  far  as  resistance  of  air  is  concerned,  ship 

steaming  at  12  knots  perpendicular  to  line  of  fire  (Col.  18) ...   84  R.  or  L. 

(c)  Calm  day,  shot  fired  in  air,  ship  steaming  at  12  knots  towards  or 

away  from  target  (Col.  14) 57  Over  or  Short 

(d)  Calm  day,  shot  fired  in  air,  ship  steaming  at  12  knots  perpen- 

dicular to  line  of  fire   (Col.  17) 70  R.  or  L. 

(e)  Ship  stationary,  12-knot  breeze  blowing  from  ship  to  target,  or  the 

reverse   (Col.  13)    27  Over  or  Short 

(f)  Ship  stationary,  12-knot  breeze  blowing  perpendicular  to  the  line 

of  fire    (Col.   16) 14  R.  or  L. 

(g)  Ship  steaming  east  at  12  knots,  real  wind  from  west  of  12  knots, 

target  abeam  to  starboard  (Cols.  16  and  17  combined) 84  Left 

(h)   Ship  steaming  east  at  12  knots,  real  wind  from  east  of  12  knots, 

target  abeam  to  starboard  (Cols.  16  and  17  combined) 56  Left 

336.  In  the  above  the  target  is  considered  as  stationary  in  every  case ;  if  it  be  not 
stationary,  then  the  errors  introduced  by  its  motion  must  be  added  algebraically  from 
Columns  15  and  18.  If  motions  be  not  in  or  perpendicular  to  the  line  of  fire,  then 
their  resolved  components  in  those  two  directions  must  be  taken. 

337.  From  what  has  already  been  said,  the  combined  effects  of  the  wind  and  of 
the  motions  of  the  firing  and  target  ships  may  therefore  be  analyzed  as  follows : 

Given  a  wind  blowing,  and  both  ship  and  target  in  motion,  there  are  really  four 
corrections  that  must  be  applied  to  correct  for  the  combined  errors  produced  by  these 
three  causes,  although  the  columns  of  the  range  tables  give  separately  corrections  for 
only  three  causes.    They  are  as  follows : 


RANGE  TABLES;  THEIE  COMPUTATIOX  AXD  USE 


203 


B. 


Correction  for  real  wind.  (The  cor- 
rections for  this  are  computed  and 
given  in  Columns  13  and  16.) 

Correction  for  the  wind  caused  by 
the  motion  of  the  gun.  (Not 
given  by  itself  in  any  columns.) 

Correction  for  the  motion  of  the  gun 
itself,  disregarding  the  effect  of 
the  wind  created  by  such  motion ; 
that  is,  the  distance  the  ship  will 
travel  during  the  time  of  flight, 
T  seconds.  (Given  in  Columns 
15  and  18.) 


These  two  corrections  if  combined  cor- 
rect for  apparent  wind,  by  using 
Columns  13  and  16. 

r  These  two  corrections  if  combined  cor- 
rect for  the  total  effect  of  the  mo- 
tion of  the  gun,  including  effect  of 
the  wind  created  by  the  motion  of 
the  gun,  by  using  Columns  14  and 
17. 


D. 


Correction  for  the  motion  of  the 
target;  that  is,  the  distance  the 
target  ship  will  travel  during  the 
time  of  flight,  T  seconds.  (Given 
in  Columns  15  and  18.) 

338.  From  the  above  tabular  statement  we  see  at  once  that  if  we  use  the  real 
wind  in  our  computations  we  must : 

(1)  Correct  for  A  by  the  use  of  Columns  13  and  16. 

(2)  Correct  for  B  and  C  combined  by  the  use  of  Columns  14  and  17. 

(3)  Correct  for  D  by  the  use  of  Columns  15  and  18. 
If  we  wish  to  use  the  apparent  wind  we  must : 

(1)  Correct  for  A  and  B  combined  by  the  use  of  Columns  13  and  16. 

(2)  Correct  for  C  by  the  use  of  Columns  15  and  18. 

(3)  Correct  for  D  by  the  use  of  Columns  15  and  18. 

339.  In  other  words,  it  is  merely  a  question  of  how  it  is  preferred  to  consider  the 
wind  effect  created  by  the  motion  of  the  gun  (B)  ;  whether  as  a  part  of  the  wind  (A), 
or  as  a  part  of  the  motion  of  the  gun  (C).  If  it  be  considered  as  a  part  of  C  the  con- 
ditions are  those  for  which  the  range  tables  are  computed  and  the  process  is  to  correct 
for: 

(1)  Real  wind  by  Columns  13  and  16. 

(2)  Motion  of  gun  by  Columns  14  and  17. 

(3)  Motion  of  target  by  Columns  15  and  IS. 

If,  however,  B  be  considered  as  a  part  of  A,  we  are  then  dealing  with  an 
apparent  wind,  and  must  not  use  Columns  14  and  17  at  all,  but  must  correct  for: 

(1)  Apparent  wind  by  Columns  13  and  16. 

(2)  Motion  of  the  gun  by  Columns  15  and  18. 

(3)  Motion  of  target  by  Columns  15  and  18. 

340.  Let  us  now  take  our  standard  problem  12"  gun,  for  10,000  yards.  If  the 
ship  be  steaming  90°  at  20  knots,  and  there  be  an  apparent  wind  blowing  from  62.5° 
at  32  knots  an  hour,  if  the  target  bears  45°  at  the  moment  of  firing,  we  have  from  the 
above  rules,  the  target  being  stationary : 


204 


EXTERIOE  BALLISTICS 


Affects. 

Formulae. 

Range. 

Deflection. 

Cause  of  error. 

Short. 
Yds. 

Over. 

Yds. 

Right. 
Yds. 

Left. 
Yds. 

Motion    of    gun.  ^ 
Cols.  15  and  18.' 

Apparent    wind. 
Cols.  13  and  16." 

Range 

Deflection.. 

Range 

Deflection.. 

ftJ.                                             ft  J. 

20  cos  45  X  ^  =  14.1  X  j^ 

20sin45x^  =  14.1X  -g 

32  cos  17.5  X  ^  =  30.5  X  ^ 

32  sin  17.5  xj^=  9.65  X  -^ 

68.625 

98.7 

98.7 

11.258 

68.625 

98.7 
68.625 

98.7 
11.258 

11.258 

■ 

30.075 
yards 
over. 

87.442 
yards 
right. 

These  are  the  combined  errors  caused  by  the  motion  of  the  gun  and  of  the  apparent 
wind. 

341.  If  we  plot  the  above  speed  of  ship  and  apparent  wind  to  scale,  which  is 
sufficiently  accurate  and  much  simpler  and  quicker  than  solving  the  oblique  triangle 
mathematically,  we  will  find  that  the  corresponding  real  wind  was  blowing  from  30° 
with  a  velocity  of  17  knots  an  hour.  Let  us  now  compute  the  results  by  using  the 
speed  of  the  ship  and  the  real  wind  by  the  methods  originally  explained,  and  com- 
pare them  with  the  results  obtained  in  the  preceding  paragraph  by  using  the 
apparent  wind.    The  work  becomes : 


Afl'ects. 

Formulae. 

Range. 

Deflection. 

Cause  of  error. 

Short. 
Yds. 

Over. 

Yds. 

Right. 
Yds. 

Left. 
Yds. 

Motion    of    gun. 
Cols.  14  and  17.' 

Real  wind.  Cols,  j 
13  and  16 ' 

Range 

Deflection.. 
Range ..... 
Deflection.. 

20  cos  45  X  ^  =  1-t-l  X  -j^ 

20  .-in  45  X  -^  =  14.1  X  ~ 

27                      27 
17  cos  15  X  ^  =  16.4  X  ^ 

17  sin  45  X  ^  =    4.4  X  -^d 

36.9 

66.975 

82.25 
5.133 

36.9 

66.975 
36.9 

87.383 
yards 
right. 

' 

30.075 
yards 
OA'er. 

The  above  are  therefore  the  errors  produced  by  the  motion  of  the  gun  and  real 
wind  combined,  and  we  see  that  the  results  are  the  same  (within  decimal  limits)  as 
those  obtained  in  the  preceding  paragraph  where  we  worked  with  the  motion  of  the 
gun  and  the  apparent  instead  of  the  real  wind. 


EANGE  TABLES;  THEIR  COMPUTATION  AND  USE 
EXAMPLES. 


205 


1.  Find  the  actual  range  for  each  gun  given  in  the  following  table  for  the  actual 
initial  velocity  given,  by  the  use  of  Column  10  of  the  range  table. 


DATA. 

ANSWERS. 

Problem. 

Gun. 

Initial  velocity. 

Range  under 

standard 

conditions. 

Yds. 

Range  for 

actual  initial 

velocity. 

Yds.' 

Cal. 
In. 

7r. 
Lbs. 

c. 

Standard, 
f.s. 

Actual, 
f.s. 

A 

3 
3 

4 

5 

5  . 

6 

6 

6 

7 

7 

8 
10 
10 
12 
13 
13 
14 
14 

13 

13 

33 

50 

50 

105 

105 

105 

165 

165 

260 

510 

510 

870 

1130 

1130 

1400 

1400 

1.00 
1.00 
0.67 
1.00 
0.61 
0.61 
1.00 
0.61 
1.00 
0.61 
0.61 
1.00 
0.61 
0.01 
1.00 
0.74 
0.70 
0.70 

1150 
2700 
2900 
3150 
3150 
2600 
2800 
2800 
2700 
2700 
2750 
2700 
2700 
2900 
2000 
2000 
2000 
2600 

1175 
2600 
2930 
3100 
3182 
2532 
2871 
2757 
2731 
2685 
2800 
2630 
2747 
2S37 
2100 
1900 
1950 
2683 

2467 

4050 

3250 

3675 

3130 

14525 

4250 

2950 

7350 

7450 

8450 

10430 

11425 

23975 

10100 

11500 

14400 

14400 

2523  5 

B 

3893  0 

C 

D 

E 

F 

3302.8 

3597.3 

3183.5 

14131  3 

Cx 

H 

4399.8 
2873  0 

I 

7451  7 

J 

7390  3 

K 

8676  0 

L 

10d92  2 

M 

11712  2 

N 

23323  0 

0 

10764  0 

P              

10692.0 

0 

13906  0 

R 

15057.4 

2.  Find  the  actual  range  of  the  guns  given  in  the  following  table  for  the  weights 
of  projectile  given,  by  the  use  of  Column  11  of  the  range  tables. 


DATA. 

ANSWERS. 

Problem. 

Gun. 

Standard 
initial  ve- 
locity, 
f.  s. 

Standard 
range. 
Yds. 

Actual 

weight  of 

projectile. 

Lbs. 

Actual  range. 
Yds. 

Cal. 
In. 

10. 

Lbs. 

c. 

C 

4 
5 

5 
6 
6 
6 
7 
7 
8 

10 
10 
12 
13 
13 
14 
14 

33 

50 

50 

105 

105 

105 

165 

165 

260 

510 

510 

870 

1130 

1130 

1400 

1400 

0.07 
1.00 
0.01 
0.61 
1.00 
D.61 
1.00 
0.61 
0.61 
1.00 
0.61 
0.61 
1.00 
0.74 
0.70 
0.70 

2900 
3150 
3150 
2000 
2800 
2800 
2700 
2700 
2750 
2700 
2700 
2900 
2000 
2000 
2000 
2600 

3100 

3600 

4000 

14600 

4050 

3600 

7000 

7000 

8100 

9000 

11100 

20800 

9000 

11300 

13500 

14200 

30 

54 

47 

110 

101 

107 

107 

160 

267 

515 

507 

875 

1125 

1135 

1413 

1391 

3199  0 

D 

3500  0 

E 

4099  0 

F 

14666  7 

G 

4114.3 

H 

3571.3 

I 

6981.4 

J \ 

7063  8 

K 

8041.2 

L 

9574.0 

M 

11116.2 

N 

20789.0 

0 

9015.9 

P.              

11288.0 

Q 

13470.1 

R 

14220.7 

206 


EXTEEIOE  BALLISTICS 


3.  Find  the  actual  range  of  the  guns  given  in  the  following  table  for  the  given 
atmospheric  conditions  by  the  use  of  Table  IV  of  the  Ballistic  Tables  and  of  Column 
12  of  the  range  tables. 


DATA. 

ANSWERS. 

Problem. 

Gun 

Standard 
initial  ve- 
locity, 
f.  s. 

Standard 

range. 

Yds. 

Atmosphere. 

Actual  range. 
Yds. 

Cal. 
In. 

10. 

Lbs. 

c. 

Bar. 
In. 

Ther. 
°F. 

A 

3 
3 

4 
5 
5 
6 
6 
6 
7 
7 
8 

10 
10 
12 
13 
13 
14 
14 

13 

13 

33 

50 

50 

105 

105 

105 

165 

165 

260 

510 

510 

870 

1130 

1130 

1400 

1400 

1.00 
1.00 
0.67 
1.00 
0.61 
0.61 
1.00 
0.61 
1.00 
0.61 
0.61 
1.00 
0.61 
0.61 
1.00 
0.74 
0.70 
0.70 

11.50 
2700 
2900 
3150 
3150 
2600 
2800 
2800 
2700 
2700 
2750 
2700 
2700 
2900 
2000 
2000 
2000 
2600 

2000 

3400 

3500 

4100 

4500 

13400 

4300 

3750 

7300 

7500 

8200 

10100 

11200 

19000 

9700 

11300 

14000 

14500 

28.10 
28.50 
29.00 
29.50 
30.00 
30.33 
30.75 
31.00 
30.50 
30.00 
29.. 50 
29.00 
28.50 
28.00 
28.25 
29.00 
30.00 
31.00 

5 

10 
15 
20 
25 
30 
35 
40 
50 
60 
70 
75 
SO 
85 
90 
95 
100 
97 

1979.6 

B 

3296 . 5 

C 

3435 . 1 

D 

3973.0 

E 

4372.4 

F 

12844.7 

G 

4176.3 

H 

3690.7 

I 

7141  9 

J 

7470.0 

K 

8250  3 

L 

10295.0 

M 

11434.1 

N 

19684.8 

0 

10000.8 

P 

11564.0 

0 

14262.6 

R 

14621.8 

4.  Find  the  errors  in  range  and  deflection  caused  by  the  real  wind  in  the  prob- 
lems given  below,  using  the  traverse  tables  and  Columns  13  and  16  of  the  range 
tables. 


DATA. 

ANSWERS. 

Problem. 

Gun. 

Ini- 
tial 
veloc- 
itv. 
f.  s. 

Range. 
Yds. 

Line 
of  fire. 
°True. 

Real  wind. 

Errors  due  to  wind. 

Cal. 
In. 

to. 
Lbs. 

c. 

From. 
°True. 

Veloc- 
ity. 
Knots. 

In  range. 

Yds. 
Short  or 

over. 

In  deflection. 

Yds. 

Right  or 

left. 

A 

3 
3 

4 
5 
5 
6 
6 
6 
7 
7 
8 

10 
10 
12 
13 
13 
14 
14 

13 

13 

33 

50 

50 

105 

105 

105 

165 

165 

260 

510 

510 

870 

1130 

1130 

1400 

1400 

1.00 
1.00 
0.67 
1.00 
0.61 
0.61 
1.00 
0.61 
1.00 
0.61 
0.61 
1.00 
0.61 
0.61 
1.00 
0.74 
0.70 
0.70 

1150 
2700 
2900 
3150 
3150 
2600 
2800 
2800 
2700 
2700 
2750 
2700 
2700 
2900 
2000 
2000 
2000 
2600 

2600 

4000 

3800 

4100 

3500 

1.3000 

3900 

4000 

7300 

6600 

8400 

10400 

11400 

24000 

10000 

11000 

14200 

14000 

35 

150 

200 

270 

300 

23 

70 

90 

225 

260 

45 

225 

70 

33 

330 

80 

350 

37 

22 

37 

45 

0 

350 

170 

280 

90 

100 

240 

180 

300 

200 

95 

115 

210 

23 

105 

15 
18 
20 
21 
17 
25 
30 
13 
20 
18 
16 
4 
23 
30 
20 
15 
27 
19 

21.9  short 

19.3  over 
19.6  over 

0.0 

7.3  short 
187.3  over 

31.4  over 
9.75  short 
44.9  over 

32 . 4  short 
28.2  over 

5.0  short 

55.5  over 
175.0  short 

87 . 1  over 

46 . 4  over 

162.0  short 

34.9  short 

2  7  rio-ht 

B 

30.4  right 

C 

5.0  right 
21  7  left 

D 

E 

4  3  left 

F 

78.2  left 

G 

10  1  rio-ht 

H 

0.0 

I 

40.2  right 

J 

K 

6.2  right 
16.0  left 

L 

12  3  left 

M 

36.6  left 

N 

194.3  left 

0 

33.6  left 

P 

28.8  left 

Q 

53.9  left 

R 

47.0  left 

EANCtE  TABLES;  THEIR  COMPUTATION  AND  USE 


207 


5.  Find  the  errors  in  range  and  deflection  caused  by  the  motion  of  the  gun  in  the 
problems  given  below,  using  traverse  tables  and  Columns  14  and  17  of  the  range 
tables. 


DATA. 

ANSWERS. 

Problem. 

Gun. 

Ini- 
tial 
veloc- 
ity. 

f.  3. 

Range. 

Yds. 

Line 
of  fire. 
°True. 

Course 
of  ship. 
°True. 

Speed 
of  ship.' 
Knots. 

Errors  due  to  speed  of 
firing  ship. 

Cal. 
In. 

10. 

Lbs. 

c. 

In  range. 

Yds." 
Short  or 

over. 

In  deflection. 

Yds. 

Right  or 

left. 

A 

B 

C 

D 

E 

F 

G 

H 

I 

J 

K 

L 

M 

N 

0 

P 

Q 

R 

3 
3 

4 
5 
5 

6 

6 

6 

7 

7 

8 

10 

10 

12 

13 

13 

14 

14 

13 

13 

33 

50 

50 

105 

105 

105 

165 

165 

260 

510 

510 

870 

1130 

1130 

1400 

1400 

1.00 
1.00 
0.67 
1.00 
0.61 
0.61 
1.00 
0.61 
1.00 
0.61 
0.61 
1.00 
0.61 
0.61 
1.00 
0.74 
0.70 
0.70 

1150 
2700 
2900 
3150 
3150 
2600 
2800 
2800 
2700 
2700 
2750 
2700 
2700 
2900 
2000 
2000 
2000 
2600 

2100 

3700 

3400 

4200 

3900 

13500 

3100 

2700 

7200 

6900 

8400 

10200 

11300 

19600 

10300 

11100 

14100 

13600 

37 
40 
200 
205 
75 
270 
145 
270 
210 
250 
300 
240 
247 
199 
275 
105 
247 
303 

0 
100 
135 
340 
280 

60 
145 

90 
120 
340 
160 

27 
215 
160 

32 
223 
330 
162 

10 
15 
20 
25 
30 
27 
25 
22 
20 
19 
17 
15 
15 
30 
20 
8 
27 
23 

20.0  over 
11.3  over 

13.5  over 

28.0  short 

43 . 1  short 
130.6  short 

38.2  over 
31.2  short 

0.0 
0.0 

53 . 1  short 

56.7  short 
69 . 0  over 

188.3  over 

57.8  short 

27.6  short 

29 . 2  over 
120.8  short 

18.6  left 

30.1  right 

36.2  left 
39.8  right 
25.4  right 

121.5  right 
0.0 
0.0 

90.3  left 

80.8  right 
56 . 3  left 

52.7  risrht 

55.9  left 
220.5  left 
157.8  right 

68.0  right 
328.3  right 
129.3  left 

6.  Find  the  errors  in  range  and  in  deflection  caused  by  the  motion  of  the  target 
in  the  problems  given  below,  using  the  traverse  tables  and  Columns  15  and  18  of  the 
range  tables ;  giving  also  the  setting  of  the  sight  in  deflection  to  compensate  for  the 
deflection. 


Problem, 


DATA. 


Gun. 


Cal. 

u\ 

In. 

Lbs. 

A 

3 

13 

B 

3 

13 

C 

4 

33 

D 

5 

50 

E 

5 

50 

F 

6 

105 

G 

6 

105 

H 

6 

105 

1 

7 

165 

J 

7 

165 

K 

8 

260 

L 

10 

510 

il 

10 

510 

N 

12 

870 

0 

13 

1130 

P 

13 

1130 

Q... 

14 

1400 

R 

14 

1400 

Ini- 
tial 
veloc- 
ity, 
f.  s. 


1150 
2700 
2900 
3150 
3150 
2600 
2800 
2800 
2700 
2700 
2750 
2700 
2700 
2900 
2000 
2000 
2000 
2600 


Range. 
Yds. 

Line 
of  fire. 
°True. 

2000 

220 

4100 

1.55 

3700 

110 

4200 

320 

4400 

242 

13700 

73 

3800 

73 

3600 

332 

7000 

143 

7400 

343 

8300 

135 

10200 

2.50 

11400 

320 

22000 

67 

10300 

313 

11500 

137 

14300 

57 

13700 

45 

Course 
of  tar- 
get. 
°True. 


Speed 
of  tar- 
get. 
Knots. 


90 

32 

157 

287 

311 

201 

73 

152 

53 

73 

0 

270 

10 

300 

47 

37 

330 

180 


35 
32 
37 
33 
30 
28 
27 
26 
25 
22 
20 
19 
18 
16 
15 
20 
23 


ANSWERS. 


Errors  due  to  motion 
of  target. 


Range. 

Yds. 
Short  or 


17 

86, 

58, 
103 

35 
288. 

81. 

67. 
0. 
0. 
101, 
177 
112, 
208, 

13. 

32. 

16. 
187. 


0  over 
4  over 

1  short 

3  short 

4  short 

3  over 
9  short 

5  over 
0 

0 

4  over 
0  short 
8  short 
8  over 


Deflection 

Yds. 

Right  or 

left. 


20 

133, 
62, 
07, 
92, 

307. 
0. 
0. 

172. 

147. 

101. 
64. 

135. 

278. 


2  over    191 
9  over    187 

3  short!.325 

4  over  |187 


3  right 
0  right 

4  left 

3  right 

4  left 

8  left 
0 

0 

0  right 

9  left 

4  right 

0  left 

1  left 

4  right 
6  left 

5  right 
0  right 
4  left 


Set  sight 
for  deflec- 
tion at 
knots  of 
scale. 


43.9 
20. d 
73.4 
29.8 
80.8 
73.6 
50.0 
50.0 
24.0 
75.0 
34.4 
56.8 
64.6 
35.6 
66.0 
35.2 
30.0 
60.3 


208 


EXTEEIOE  BALLISTICS 


7.  Given  the  apparent  wind,  tlie  motions  of  the  gun  and  target,  and  the  actual 
range  and  bearing  of  the  target  from  the  gun,  as  shown  in  the  following  table,  com- 
pute the  errors  in  range  and  in  deflection  resulting  from  those  causes,  and  tell  how 
to  set  the  sights  in  range  and  in  deflection  in  order  to  hit. 


DATA. 


£ 

Gun. 

Initial 

Actual 

Bearing 

of 
target. 
"True. 

Gun. 

Target. 

Apparent  wind. 

0) 

Cal. 

w. 

veloc- 
ity. 
f.  s. 

range 
Yds. 

Course. 

Speed. 

Course. 

Speed. 

From. 

Veloc- 
ity. 
Knots. 

u 

In. 

Lbs. 

°True. 

Knots. 

°True. 

Knots. 

°True. 

A  . 

3 

13 

1.00 

1150 

2300 

15 

45 

6 

80 

10 

180 

10 

B  . 

3 

13 

1.00 

2700 

3800 

260 

315 

35 

200 

30 

290 

52 

C 

4 

33 

0.67 

2900 

3400 

45 

220 

22 

260 

25 

48 

15 

D. 

5 

50 

1.00 

3150 

4000 

153 

1.53 

25 

153 

30 

300 

25 

E  . 

5 

50 

0.61 

3150 

4100 

75 

67 

25 

80 

32 

10 

42 

F.. 

6 

105 

0.61 

2600 

11600 

300 

110 

20 

130 

25 

270 

15 

G. 

6 

105 

1.00 

2800 

4000 

45 

180 

20 

90 

20 

315 

18 

H. 

6 

105 

0.61 

2800 

3400 

27 

305 

22 

350 

24 

120 

10 

1.. 

7 

165 

1.00 

2700 

6600 

265 

37 

18 

190 

19 

15 

37 

J.. 

7 

165 

0.61 

2700 

6300 

170 

135 

20 

170 

35 

340 

12 

K  . 

8 

260 

0.61 

2750 

8100 

200 

220 

21 

315 

30 

300 

30 

L.. 

10 

510 

1.00 

2700 

9700 

110 

275 

15 

275 

20 

52 

20 

M. 

10 

510 

0.61 

2700 

10700 

270 

330 

18 

330 

25 

190 

35 

N. 

12 

870 

0.61 

2900 

20600 

22 

15 

22 

30 

20 

330 

45 

0.. 

13 

11,30 

1.00 

2000 

10200 

345 

103 

15 

120 

20 

355 

15 

P.. 

13 

11.30 

0.74 

2000 

10900 

60 

227 

12 

80 

15 

240 

30 

Q.. 

14 

1400 

0.70 

2000 

13800 

320 

340 

21 

165 

19 

300 

25 

R.. 

14 

1400 

0.70 

2600 

14200 

125 

17 

21 

23 

17 

325 

20 

ANSWERS. 


Combined  errors 

Set  sights  at — 

Problem. 

Range. 

Yds. 

Short  or   over. 

Deflection. 

Yds. 

Right  or  left. 

Range. 
Yds. 

Deflection. 
Knots. 

A 

16.0  over 
91.2  short 
15.9  short 
19.6  over 
37.2  short 
32.8  short 

88.2  short 

37.3  short 

62.4  short 

69.1  short 

62.5  short 
4.8  short 

50.1  short 

220.9  short 

2.2  over 

162.9  short 

413.4  over 

59.4  over 

25.5  left 
178.1  right 

38.8  right 
13.3  left 

1.8  right 
1.30.0  riglit 

12.9  right 
19.8  left 

125.8  right 

57.5  left 
262.3  left 

34.6  right 

11.3  right 
95.1  right 
18.5  left 

28.4  left 
265.7  right 

22.4  left 

2284.0 

3891.2 

3415.9 

3980.4 

4137.2 

11632.8 

4088.2 

3437.3 

6662.4 

6369.1 

8162.5 

9704.8 

10750.1 

20820.9 

10197.8 

11062.9 

13386.6 

14140.6 

56.5 

B 

6.0 

c 

34.0 

D 

r,4.3 

E 

49.4 

F 

G 

39.4 
45.9 

H 

58.5 

I 

29.3 

J 

61.9 

K 

92.0 

L 

46.0 

M 

48.7 

N 

44.6 

0 

51.6 

P 

52.4 

0 

32.9 

R 

51.9 

EANGE  TABLES;  THEIE  COMPUTATION  AND  USE 


209 


8.  Given  the  data  contained  in  the  following  table,  find  how  the  sights  must  be 
set  in  range  and  deflection  in  order  to  hit.  Use  traverse  tables  for  all  resolutions  of 
forces. 


C. 

E. 

H. 

J. 

K. 

M. 

N. 

P. 

R. 

Caliber  of  gun.  . .  . 

4 

5 

6 

7 

8 

10 

12 

13 

14 

Standard    initial 

velocity 

2900 

3150 

2800 

2700 

2750 

2700 

2900 

2000 

2600 

Standard    weight 

of  projectile  .... 

33 

50 

105 

163 

260 

510 

870 

1130 

1400 

CoetHcient  of  form. 

0.67 

0.61 

O.Gl 

0.61 

0.61 

0.61 

0.61 

0.74 

0.70 

Course     of    firing 

ship,  °true 

90 

300 

75 

200 

25 

330 

230 

115 

27 

Speedof  tiring  ship, 

knots 

13 

19 

17 

25 

IS 

22 

20 

IS 

15 

Course  of  enemy, 

°true  

180 

200 

110 

200 

25 

GO 

10 

70 

330 

Speed    of    enemy, 

knots 

28 

21 

19 

20 

30 

20 

15 

22 

IS 

Bearing  of  enemy 

at  moment  of  fir- 

ing, °true 

135 

260 

165 

20 

25 

330 

90 

80 

70 

Distance  of  enemy 

at  moment  of  fir- 

ing, yards 

3400 

4300 

4000 

7500 

7000 

10500 

20000 

11000 

14500 

Direction    from 

which  real  wind 

is  blowing,  °true. 

315 

25 

300 

223 

90 

150 

330 

200 

10 

Velocity    of   real 

wind,  knots 

30 

19 

25 

18 

16 

21 

18 

20 

13 

Barometer,  inches. 

30.70 

30.00 

30.50 

30.25 

30.00 

29.50 

29.00 

29.33 

30.15 

Temperature    of 

air,  °F 

50 

25 

75 

80 

30 

15 

80 

90 

40 

Temperature    of 

powder,  °F 

80 

65 

93 

97 

75 

70 

93 

100 

60 

Actual    weight  of 

projectile 

30 

32 

107 

160 

267 

500 

877 

1122 

1415 

ANSWERS. 

Total  error  in 

range,  yards.  . .  . 

10. Of 

356. 3§ 

6.6§ 

272.lt 

496. 0§ 

427.4  § 

487.lt 

264.6 t 

1123.8  § 

Total  error  in  de- 

flection, yards.. . 

69.1* 

71.5$ 

4.3* 

10.  11: 

13.3* 

166.7* 

483. 6t 

100.2$ 

160.8$ 

Exact    setting    in 

range,  yards. ..  . 

3390.0 

4656 . 3 

4006.6 

7227.9 

7496.0 

10927.4 

19512.9 

10733.4 

15625.8 

Exact    setting    in 

deflection,  knots. 

78.6 

25.5 

51.5 

48.3 

52.5 

70.0 

21.6 

41.6 

37.1 

Actual   setting  in 

range,  yards.  ..  . 

3350 

4650 

4000 

7200 

7430  or 

10900 

19500 

10700 

15600 

Actual  setting   in 

7500 

deflection,  knots. 

79 

25  or  26 

51  or  52 

48 

52  or  53 

70 

22 

42 

37 

*^left,  t  =  over,  $=z  right,  §  =  short. 


14 


210 


EXTEEIOE  BALLISTICS 


9.  Given  the  data  contained  in  the  following  tables,  find  how  the  sights  should  be 
set  to  hit.    Use  traverse  tables  for  all  resolutions  of  forces. 


C. 

D. 

F. 

I. 

K. 

L. 

N. 

0. 

Q. 

Caliber  of  gun.  . .  . 

4 

5 

6 

7 

8 

10 

12 

13 

14 

Standard    initial 

velocity 

2900 

3150 

2600 

2700 

2750 

2700 

2900 

2000 

2000 

Standard    weight 

of  projectile.  ..  . 

33 

50 

105 

165 

260 

510 

870 

1130 

1400 

Coefficient  of  form. 

0.67 

1.00 

0.61 

1.00 

0.61 

1.00 

0.61 

1.00 

0.70 

Course  of  firing 

ship,  °true 

33 

115 

213 

302 

350 

265 

171 

105 

77 

Speed    of   firing 

ship,  knots 

35 

27 

25 

20 

20 

19 

16 

18 

23 

Course  of  enemy, 

°true 

357 

307 

245 

45 

135 

0 

180 

81 

349 

Speed    of    enemy, 

knots 

32 

30 

28 

15 

20 

21 

23 

25 

19 

Bearing  of  enemy 

at   moment    of 

firing,  °true 

67 

345 

23 

180 

287 

111 

351 

265 

223 

Distance  of  enemy 

at    moment    o  f 

firing,  yards 

3600 

3400 

13000 

6800 

7700 

9300 

16200 

9700 

14000 

Direction     from 

which  apparent 

wind  is  blowing. 

^ 

°true 

21 

97 

165 

237 

300 

7 

214 

165 

107 

Apparent  velocity 

of  wind,  knots.  . 

52 

43 

38 

28 

27 

5 

25 

22 

30 

Barometer,  inches. 

•28.00 

29.00 

30.00 

31.00 

30.00 

29.00 

29.00 

28.00 

30.00 

Temperature      o  f 

air,  °F 

15 

40 

50 

60 

70 

80 

90 

85 

95 

Temperature      o  f 

powder,  ^F 

60 

70 

75 

80 

85 

95 

99 

97 

100 

Actual  weight   of 

projectile 

35 

48 

107 

162 

208 

507 

876 

1123 

1408 

ANSWERS. 


Total  error  in 
range,  yards.  . .  . 

Total  error  in  de- 
flection, yards..  . 

Exact  setting  in 
range,  yards. . .  . 

Exact  setting  in 
deflection,  knots. 

Actual  setting  in 
range,  yards.  . .  . 

Actual  setting  in 
deflection,  knots. 


204.1  § 

45.9$ 

3804.1 

32.2 

3800 

32 


159. 2§ 

71.2$ 

3559.2 

21.5 

3550 

21  or  22 


265. 5§ 

73.0$ 

13265.5 

44.9 

13250 

45 


261. 3§ 

126.5$ 

7061.3 

30.1 

7050 

30 


30.1  § 

151.6$ 

7730.1 

24.0 

7700 

24 


219. 8t 

242.3$ 

9080.2 

20.6 

9050 

21 


829.6 t 

101.2$ 

1.5370.4 

42.1 

15350 

42 


582.9 t 

28.0* 

9117.1 

52.5 

9100 

52  or  53 


366.7 t 

351.3* 

13633.3 

72.2 

13600 

72 


*^left,  t=zover,  $  =  right,  §^  short. 


EAJsTtE  TABLES;  THEIR  COMPUTATION  AND  USE  211 

10.  A  ship  steaming  on  a  course  jST.  W.  (p.  c.)  at  20  knots,  wishes  to  fire  a  14" 
gun  (7  =  2000  f.  s.,  iv  =  14:00  pounds,  c  =  0,70)  at  a  target  ship  which  bears  off  the 
port  bow  of  the  firing  ship  and  is  steaming  N.  E.  (p.  c.)  at  18  knots.  The  gun  is  to 
be  fired  at  the  moment  when  the  firing  ship  and  target  ship  are  12,010  and  6380  yards, 
respectively,  from  the  point  of  intersection  of  tlieir  two  courses.  A  real  wind  is  blow- 
ing from  100°  (p.  c.)  with  a  velocity  of  30  knots.  The  deviation  is  3°  W.  The 
barometer  is  at  28.13"  and  the  thermometer  at  90.5°  F.  The  temperature  of  the 
powder  charge  is  99°  F.  The  powder  has  suffered  a  loss  of  volatiles  which  increases 
the  initial  velocity  22  f.  s.,  and  has  become  damp  enough  to  reduce  the  initial  velocity 
11  f.  s.  The  actual  weight  of  the  projectile  is  1385  pounds.  The  sight  is  out  of 
adjustment  an  amount  which  is  known  to  cause  the  shot,  at  the  range  given  by  the 
above  conditions,  to  strike  25  feet  above  the  point  aimed  at  and  50  yards  to  the  left  of 
it.  How  must  the  sight  be  set  in  order  that  the  shot  may  strike  the  point  aimed  at 
on  the  enemy's  hull  ? 

Ajisivers.     Total  errors Eange  1216.4  yards  over;  deflection  193.9  yards  left. 

Exact  setting Eange  12383.6  yards;  deflection  62.6  knots. 

Actual  setting .  .  ..Eange  12350  yards;  deflection  63  knots. 

11.  A  ship  steaming  on  a  course  293°  true,  at  17  knots,  wishes  to  fire  a  6"  gun 
(7  =  2600  f.  s.,  w  =  10d  pounds,  c  =  0.61)  at  a  torpedo-boat  bearing  300°  true  and 
distant  9132  yards  at  the  moment  of  firing.  The  torpedo-boat  is  steaming  on  a 
course  320°  true,  at  35  knots.  The  barometer  is  at  29.00"  and  the  thermometer  at 
10°  F.  The  temperature  of  the  powder  is  50°  F.  The  shell  weighs  109  pounds. 
A  real  wind  is  blowing  from  74°  true  at  21  knots.  How  must  the  sights  be  set  to  hit  ? 
All  work  must  be  correct  to  two  decimal  places. 

Atisivers.     Total  errors Eange  1104.1  yards  short;  deflection  157.8  yards  left. 

Exact  setting.  ..  .Eange  10236.1  yards;  deflection  68.2  knots. 
Actual  setting  . .  .Eange  10200  yards;  deflection  68  knots. 

12.  A  ship  steaming  0°  true  at  18  knots,  desires  to  fire  a  6"  gun  (7  =  2600  f.  s., 
w  =  105  pounds,  c  =  0.61)  at  a  torpedo-boat  distant  13,600  yards,  and  bearing  90° 
true,  and  steaming  on  a  course  0°  true  at  32  knots.  A  real  wind  is  blowing  from 
180°  true  with  a  velocity  of  20  knots.  The  barometer  is  at  29.00"  and  the  thermom- 
eter at  80°  F.  The  temperature  of  the  powder  is  80°  F.  The  shell  weighs  110 
pounds.    How  must  the  sights  be  set  to  hit? 

Ansivers.     Total  errors Eange  225.2  yards  over;  deflection  203.1  yards  right. 

Exact  setting .  . .  .Eange  13374.8  yards;  deflection  36.8  knots. 
Actual  setting  ...  Eange  13350  yards;  deflection  37  knots. 


PART  V. 

THE  CALIBRATION  OF  SINGLE  GUNS  AND  OF  A 
SHIP'S  BATTERY. 

INTRODUCTION  TO  PART  V. 

The  calibration  of  a  ship's  battery  means,  in  brief,  the  process  of  adjusting  the 
sights  so  that  all  the  guns  of  the  same  caliber  will  shoot  together  when  the  sights 
are  set  alike,  and  so  that  the  salvos  will  therefore  be  well  bunched.  Formerly  it  was 
considered  necessary  for  every  ship  to  calibrate  her  battery  upon  first  going  into  com- 
mission, but  now  we  find  the  work  of  manufacture  and  installation  is  ordinarily  so 
well  done  that  calibration  practice  is  not  considered  necessary  unless  there  are 
indications  to  the  contrary.  If  the  guns  persistently  scatter  their  salvos,  and  the 
reason  for  such  a  performance  is  not  apparent,  then  it  may  become  necessary  to 
calibrate  the  battery;  and,  in  any  event,  this  form  of  test  is  so  clearly  illustrative  of 
the  principles  involved  in  directing  gun  fire,  that  it  should  be  thoroughly  understood 
by  every  naval  officer.    Part  V  deals  with  this  subject. 


•1 


CHAPTEE  18. 

THE  CAIIBEATION  OF  A  SINGLE  GUN. 

New  Symbols  Introduced. 


'  Angles  for  plotting  the  point  of  fall  of  the  shot. 
Coordinates  of  point  of  fall  of  shot. 


y- 

a,  a  . 

I,  v. 

c,  c' . 

d,  d' . 
h.  .  .  . Height  of  center  of  bull's  eye  above  water. 

Sh.  ■  ■  .Danger  space  for  height  h. 

342.  Calibration  may  be  defined  as  the  process  of  firing  singly  each  gun  of  a   Definition, 
ship's  batter}',  noting  carefully  the  position  of  the  point  of  fall  of  each  shot,  finding 

the  average  point  of  fall  for  each  gun  of  the  battery,  and  then  adjusting  the  sight 
scales  of  each  gun  so  that  all  the  shots  fired  from  all  the  guns  will  fall  at  the  same 
average  distance  from  the  ship  when  all  sight  bars  are  set  to  the  same  reading;  and 
similarly  in  deflection.* 

343.  As  will  be  realized  from  what  follows,  the  process  of  holding  a  calibration 
practice  involves  considerable  time,  labor  and  expense.  It  is  evident  that,  if  accurate 
salvo  firing  is  to  be  carried  on,  all  sight  scales  must  be  so  adjusted  that,  if  the  guns  be 
properly  pointed,  the  shot  from  them  will  all  fall  well  bunched,  if  the  sights  be  all  set 
alike ;  and  the  method  of  adjusting  the  sights  to  accomplish  this  by  holding  a  calibra- 
tion practice  will  now  be  described,  as  illuminative  of  the  principles  involved. 

344.  Like  human  beings,  and  like  all  other  kinds  of  machinery,  each  particular  Peculiarities 

°  "^  .      of   individual 

gun  has  its  own  individual  peculiarities.  With  its  own  sights  properly  adjusted  it  guns, 
may  shoot  consistently  and  regiilarly,  the  shot  falling  in  a  well-grouped  bunch;  but 
this  bunch  may  not  coincide  with  the  ])unch  of  shot  fired  from  another  gun  of  the 
same  battery  in  another  part  of  the  ship,  even  if  the  latter  also  has  its  sights 
theoretically  perfectly  adjusted,  and  set  for  the  same  range  and  deflection  as  the  first. 
Also  the  bunches  of  shot  from  these  two  guns  may  neither  of  them  coincide  with 
that  from  still  another  gun.  Thus  each  gun  may  work  well  individually,  and  yet 
the  battery  may  not  be  doing  proper  team  work.  And  yet,  with  our  modern  method 
of  fire  control  and  firing  by  salvos,  it  is  of  the  utmost  importance  that,  when  the 
spotter  causes  all  the  sights  of  a  battery  to  be  set  alike,  the  shots  of  all  the  guns,  if 
fired  together,  shall  strike  as  nearly  together  as  the  inherent  errors  of  the  guns 
themselves  will  permit.  It  is  to  accomplish  this  that  calibration  practice  is  held.  It 
must  be  noted  that,  in  this  discussion,  it  is  supposed  that  all  preventable  errors  in 
pointing,  etc.,  have  been  eliminated;  nothing  can  be  well  done  as  long  as  any  of 
these  remain. 

*  If  there  be  any  question  in  the  mind  of  the  student  as  to  the  meaning  of  any  of  the 
terms  used  in  this  chapter  relative  to  the  mean  point  of  impact,  deviations,  deflections,  etc., 
reference  should  be  made  to  the  definitions  given  in  the  opening  paragraphs  of  Chapter  20; 
and  those  definitions  are  to  be  considered  as  included  in  the  lesson  covering  the  present 
chapter  so  far  as  they  may  be  necessary  to  a  clear  understanding  of  the  subject  of  cali- 
bration. 


216 


EXTEEIOE  BALLISTICS 


Mean  point 
of  impact. 


Mean 
dispersion. 


Calibration 
range. 


345.  There  are  many  causes  which  may  operate  to  produce  the  condition  in 
which  one  well-adjusted  and  well-pointed  gun  lands  its  shot  at  a  point  quite  widely 
separated  from  the  point  of  fall  of  the  shot  from  another  equally  well-adjusted  and 
well-pointed  gun ;  as,  for  example,  the  fact  that  there  is  more  give  to  the  deck  under 
one  of  the  guns  than  under  the  other,  etc. 

346.  If  a  great  number  of  shot  be  fired  from  a  gun,  under  as  nearly  as  possible 
the  same  conditions,  it  will  be  found  that  the  impacts  are  grouped  closely  together 
around  one  point,  which  point  we  will  call  the  "  mean  point  of  impact."  This  point 
is  in  reality  the  mathematical  center  of  gravity  of  all  the  impacts ;  and,  with  refer- 
ence to  the  target  (at  the  range  for  which  the  gun  is  pointed)  this  mean  point  of 
impact  may  be  either  on,  short  of,  or  over  the. target;  and  either  on,  to  the  right, 
or  to  the  left  of  the  target. 

347.  The  point  of  fall  of  each  individual  shot  is  situated  at  a  greater  or  less 
distance  from  the  mean  point  of  impact;  and  the  arithmetical  average  of  these 
distances  from  the  mean  point  of  impact  for  all  the  shots  from  the  gun  is  called  the 
*'  mean  deviation  from  the  mean  point  of  impact  *''  or  the  "  mean  dispersion  "  of  the 

3q , G>A 


Figure  32. 

gun.  In  dealing  with  these  quantities  it  is  customary  to  consider  deviations  or  errors 
in  range  separately  from  those  in  deflection ;  so  we  would  speak  of  the  "  mean 
deviation  (or  error)  in  range  from  the  mean  point  of  impact,"  and  similarly  for 
deflection.  These  quantities  are  also  called  the  "  mean  errors  "  of  the  gun  in  range 
and  in  deflection. 

348.  The  general  plan  of  a  calibration  range  is  shown  in  Figure  32.  A  raft 
carrying  a  vertical  target  screen  is  moored  in  such  a  position  that  one  or  more 
observing  stations,  preferably  two,  may  be  established  on  shore,  as  shown  at  A  and 
B.  The  ship  is  then  moored  at  S,  broadside  to  the  target;  and  the  screen  of  the 
latter  should  be  as  nearly  as  possible  parallel  to  the  keel  of  the  ship.  The  angle  STA 
should  be  as  nearly  a  right  angle  as  possible. 

349.  The  base  line  AB  must  then  be  measured  or  determined  by  surveying 
methods;  and  then  the  positions  of  the  ship,  target,  etc.,  must  be  accurately  plotted, 
and  their  distance  apart  accurately  determined. 


CALIBRATIOX  OF  SIXGLE  GUXS  AXD  A  SHIP'S  BATTEEY     217 

350.  Having  found  the  distance,  »ST,  from  the  gun  to  the  target,  the  sights  of 
the  gun  to  be  calibrated  are  set  to  that  range,  and  a  string  (usually  of  four)  care- 
fully aimed  shots  is  fired  at  the  target.  (It  is  usual  to  set  the  sights  a  small 
known  distance  off  in  deflection,  to  prevent  damage  to  the  target  and  consequent 
frequent  delays  in  completing  practice.)  Let  us  suppose  that  the  first  shot  fell  at 
F.  By  the  use  of  sextants,  plane  tables  or  their  equivalents  (preferably  plane 
tables),  the  angles  a,  /?  and  y  should  be  observed,  and  the  point  of  fall  should  be 
plotted  on  the  drawing  board.  This  process  should  be  repeated  for  each  shot,  and  the 
results  tabulated  for  the  gun,  the  errors  in  range  and  in  deflection  being  measured 
from  the  drawing  board  for  each  shot.  This  process  is  repeated  for  each  gun  of  the 
battery,  and  in  doing  this  it  is  well  to  fire  one  shot  from  each  gun  in  turn  instead  of 
having  one  gun  fire  its  whole  allowance  at  once,  as  more  uniform  conditions  for 
firing  the  battery  as  a  whole  are  obtained  in  this  way,  especially  in  regard  to  the 
temperature  of  the  guns.  A  gun  is  not  loaded  until  immediately  before  it  is  fired,  for 
a  number  of  reasons,  among  which  is  the  fact  that  otherwise  the  temperature  of  the 
charge  would  be  changed  by  contact  with  the  heated  walls  of  the  powder  chamber. 

351.  It  is  most  desirable  in  calibration  practice  that  the  conditions  of  weather  weather 
should  be  good,  and  should  be  uniform  for  the  firing  of  all  guns  of  the  same  caliber. 

The  weather  should  be  uniform  for  the  whole  firing,  if  practicable.  If  the  weather 
be  not  uniform  throughout  the  firing  for  one  caliber,  then  it  is  necessary  that  the 
data  for  each  shot  be  reduced  to  standard  conditions  individually  before  any  com- 
bination of  the  results  of  different  shots  is  made.  The  complete  practice  should  be 
finished  in  one  day  if  possible ;  as  it  is  bad  practice  to  have  part  of  the  firing  on  the 
afternoon  of  one  day  and  the  remainder  on  the  forenoon  of  the  next,  for  instance,  as 
the  weather  conditions  may  be  entirely  different  on  the  two  days,  and  misleading 
results  may  follow  such  a  course. 

352.  The  greater  the  number  of  shots  fired  the  more  reliable  are  the  results.   Number 
Four  shots  are  usually  fired  from  each  gun,  which  is  a  small  number ;  but  the  cost  of 

the  ammunition  so  expended  is  not  small  and  limits  the  practice  to  that  allowed  by 
a  reasonable  economy,  to  say  nothing  of  the  wear  on  the  guns,  especially  on  those 
of  large  caliber. 

353.  During  the  practice,  for  each  shot,  the  observers  at  each  shore  station   observations, 
should  record: 

(a)  The  time  of  flash. 

(b)  The  consecutive  number  of  the  shot. 

(c)  The  angle  between  the  point  of  fall  and  the  center  of  the  target. 

(d)  The  force  and  direction  of  the  wind. 

354.  The  observers  on  board  ship^  in  addition  to  the  above,  should  record  the 
following  for  each  shot : 

(e)  Time  of  shot  (in  place  of  time  of  flash). 

(f)  Consecutive  number  of  shot  (should  be  same  as  (b)). 

(g)  Xumber  of  gun  from  which  fired. 

(h)  Whether  or  not  the  cross  wires  of  the  telescope  Mere  on  the  center  of  the 
bull's  eye  at  the  instant  the  gun  was  fired ;  and,  if  not,  how  much  they  were  off  in 
each  direction  (estimated  in  feet  on  the  target  screen ;  lines  painted  on  the  screen 
should  assist  in  making  this  estimate). 

(i)   Direction  and  force  of  the  wind  in  knots  per  hour. 

(j)   Barometer. 

(k)   Temperature  of  the  air. 

(1)  Temperature  of  the  charge  (assumed  as  the  same  as  the  temperature  of  the 
magazine,  from  which  the  charge  should  not  be  removed  until  it  is  actually  needed 
for  the  firing). 


218 


EXTERIOR  BALLISTICS 


Necessity 
for  care. 


Plotting  of 

observed 

points  of 

fall. 


(m)  Weight  of  the  shell. 

(n)   Any  other  information  that  may  be  desirable. 
( (i),  (j)  and  (k)  need  only  be  recorded  when  a  change  occurs,  but  the  record  must  be 
such  that  the  conditions  at  the  beginning  and  at  the  end  of  the  practice,  and  at  the 
moment  when  any  individual  shot  is  fired  may  be  readily  and  accurately  obtained 
from  it.) 

355.  The  members  of  the  observing  parties  should  realize  the  necessity  for 
accurate  observations  and  records.  Xothing  is  more  disastrous  than  carelessness 
in  regard  to  details,  as  inaccuracy  in  any  one  of  the  apparently  minor  points  may 
easily  result  in  rendering  the  results  of  the  whole  practice  entirely  worthless.  Such 
inaccuracies  may  readily  be  of  such  a  nature  that  they  cannot  be  detected,  and  might 
lead  to  confident  entry  into  battle  or  target  practice  with  a  battery  with  which  it  is 
impossible  to  do  good  work  owing  to  the  undiscovered  carelessness  or  inaccuracy  of 
some  person  charged  with  some  of  the  duties  in  regard  to  the  observations  taken 
during  the  calibration  practice. 

356.  Suppose  we  have  four  shots  fired  from  a  single  gun,  which  fell  as  follows 
relative  to  the  foot  of  the  perpendicular  from  the  center  of  the  bull's  eye  upon  the 
water : 

No.  1 a  yards  over a'  yards  to  the  right. 

No.  2 h  yards  short h'  yards  to  the  left. 

No.  3 c  yards  over c'  yards  to  the  right. 

No.  4 d  yards  short d'  yards  to  the  left. 


Figure  33 


357.  Then  their  points  of  fall  are  as  shown  in  Figure  33,  in  which  we  have  given 
a  projection  in  the  vertical  plane  through  the  line  of  fire  and  the  center  of  the  bull's 
eye,  and  also  the  corresponding  projection  upon  the  horizontal  plane  of  the  water. 
T  is  the  target,  the  center  of  the  bull's  eye  being  at  B,  which  is  li  feet  above  the  water. 
LU  is  the  line  of  sight  such  that  the  gun  pointer  looking  along  it  sees  the  cross 
wires  of  the  telescope  on  the  center  of  the  bull's  eye.    Now  if  the  gun  were  in  perfect 


CALTBEATIOX  OF  SIXGLE  GUXS  AND  A  SHIP'S  BATTEEY     219 

adjustment  when  fired,  its  sliot  would  travel  along  the  trajectory  X,  pierce  the  bull's 
eye  at  B  and  strike  the  water  at  P.  Note  that  the  recorded  errors  are  actually  observed 
from  the  point  B'  on  the  surface  of  the  water  vertically  below  B.  Therefore  we  have 
to  reduce  our  observations  to  the  point  P,  by  subtracting  for  overs  in  range,  the  dis- 
tance B'P  from  the  recorded  range,  and  by  adding  the  same  distance  for  shorts ;  and 
B'P  may  therefore  be  considered  as  a  constant  error  affecting  all  shots  alike. 

358.  The  distance  Sn  is  the  danger  space  for  a  target  h  feet  high  at  a  range 
equal  to  the  distance  from  the  gun  to  the  target  plus  the  danger  space.  For  practical 
purposes,  however,  when  the  range  is  considerable,  this  danger  space  may  be  taken 
from  the  range  tables  for  the  height  h  for  a  range  equal  to  the  distance  from  the 
gun  to  the  target.  The  amount  of  correction  to  be  applied  because  of  the  height  h 
should  also  be  taken  from  Column  19  of  the  range  table.  It  sometimes  happens, 
also,  that  the  point  of  sight  may  not  be  exactly  on  the  center  of  the  bull's  eye  at 
the  moment  of  firing,  but  may,  by  the  check  telescope,  be  determined  to  have  been  a 
certain  distance  above  or  below  the  proper  point  of  aim ;  in  which  case  h  would  have 
to  be  modified  accordingly.    To  work  out  the  observed  data : 

1.  Take  a  large  drawing  board,  and  plot  on  it  to  scale  (scale  sufficiently  large 
to  give  accurate  results)  the  positions  of  the  observing  parties,  gun  and  center  of 
the  target. 

2.  Using  the  observed  data,  plot  the  point  of  fall  of  each  shot,  and  measure  the 
distance  from  the  foot  of  the  perpendicular  through  the  center  of  the  bull's  eye 
on  the  water,  in  range  and  in  deflection,  recording  the  results. 

3.  From  the  results  obtained  by  (2),  find  the  location  of  the  mean  point  of 
impact  in  range  and  in  deflection  with  reference  to  the  perpendicular  noted  in  (2), 

which  for  range  will  be  ~ — --^ ^  ^ — -^  yards  from  the  foot  of  the  perpen- 

dicular; a,  h,  c  and  d  being  taken  with  their  proper  algebraic  signs,  -f-  for  an  over 
and  —  for  a  short;  the  sign  +  on  the  result  showing  that  the  point  is  over  and  — 
that  it  is  short. 

4.  For  the  mean  point  of  impact  in  deflection,  by  similar  methods,  the  distance 

from  the  line  of  sight  will  be  +o,' +  i-^')+c' +  {-d')  .  ^,^  ^,^  ^,  ^^^^  ^,  ^^^.^^  ^^j.^^ 

4 
with  their  proper  algebraic  signs,  4-  for  a  deviation  to  the  right  and  —  to  the  left ;  a 
4-  sign  on  the  result  will  show  that  the  mean  point  of  impact  is  to  the  right  of  the  line 
of  fire  and  a  —  sign  that  it  is  to  the  left.  It  is  equally  as  good,  and  sometimes  more 
convenient,  instead  of  using  the  4-  and  —  signs  in  this  work,  to  keep  to  the  nomen- 
clature of  "  short  "  and  "  over,"  etc. ;  using  the  letters  "  S  "  and  "  0  "  to  represent 
them  and  "  K  "  and  '"  L  "  to  represent  "  rights  "  and  "  lefts  ";  thus  a  shot  might  be 
"  155  S  and  25  L." 

359.  Having  obtained  from  these  plotted  positions  for  the  particular  group  of  Reduction 
shots  under  consideration  the  mean  distances  in  range  and  in  defiection  from  the  foot  conditions, 
of  the  perpendicular  on  the  water  through  the  bull's  eye,  it  is  now  necessary  to  reduce 

those  distances,  first  to  the  point  P  as  an  origin,  and  then  from  firing  to  standard  con- 
ditions. The  method  of  doing  this  is  simply  an  application  of  methods  that  have 
already  been  studied  in  this  book,  and  it  may  be  best  understood  from  the  solution  of  a 
problem. 


220 


EXTEEIOR  BALLISTICS 


Calibration 
problem. 


360.  Let  US  suppose  that  six  shots  were  fired  on  a  calibration  practice  from  a 
12"  gun  (y  =  2900  f.  s.,  w  =  870  pounds,  c  =  0.61)  under  the  following  conditions: 

Actual  distance  of  target  from  gun 8000  yards. 

Sights  set  in  range  for 8000  yards. 

Sights  set  in  deflection  at 38  knots. 

Center  of  bull's  eye  above  the  water 12  feet. 

Bearing  of  target  from  ship -45°  true. 

Wind   blowing  from 270°  true. 

Wind  blowing  with  a  velocity  of 18  knots. 

Barometer    30.00". 

Temperature  of  the  air 75°  F. 

Temperature  of  the  powder 94°  F. 

Weight   of   projectile 875  pounds. 

Measured  from  the  foot  of  the  perpendicular  upon  the  water  through  the  center  of 
the  bull's  eye,  the  shot  fell. 

'No.  1.  .200  yards  short;  90  yards  left.  No.  4.  .150  yards  short;  85  yards  left. 
No.  2.  .150  yards  short;  95  yards  left.  No.  5.  .100  yards  short;  75  yards  left. 
No.  3.  .100  yards  short;  95  yards  left.       No.  6.  .    50  yards  short;  70  yards  left. 

Find  the  true  mean  errors  in  range  and  in  deflection  under  standard  conditions,  and 
adjust  the  sight  scales  in  range  and  in  deflection  in  order  to  have  the  sights  properly 
set ;  that  is,  under  standard  conditions,  to  have  the  mean  point  of  impact  at  the  point 
P  when  the  sight  is  set  for  8000  yards  in  range  and  for  50  knots  on  the  deflection  scale. 


No.  of  shot. 

Range, 
Short. 

Yds. 

Deflection. 
Left. 
Yds. 

1 

200 
150 
100 
150 
100 
50 

90 

2 

95 

3 

95 

4 

85 

5 

75 

6 

70 

Mean  errors  on  foot  of    perpendicular") 
through  bulTs  eye.                                  J 

6|750 
125 
yards 
short. 

6|510 

85 
vards 

"left. 

The  error  in  range  due  to  the  fact  that  the  point  of  aim  is  at  the  bull's  eye  and 
not  at  the  water  line  of  the  target  is  the  correction  that  should  be  applied  to  the 
observed  distance  from  the  foot  of  the  perpendicular  on  the  water  through  the  bull's 
eye  in  order  to  refer  it  to  the  point  P  as  an  origin.    By  Column  19  of  the  range  tables, 

this  would  be  12  X  -^  =  60  yards. 

The  error  in  deflection  intentionally  introduced  in  order  to  avoid  wrecking  the 
target,  by  setting  the  sight  off  in  deflection,  would  be,  by  Column  18  of  the  range  table, 

(50-38)  X  "I  =65  yards  left. 

Now  to  bring  the  observed  errors  to  their  true  values  under  standard  conditions, 
we  proceed  as  follows : 


CALIBEATION  OF  SINGLE  GUNS  AND  A  SHIP'S  BATTERY     221 


Temperature  of  the  powder  is  4°   above  the  standard,  therefore  the  initial 

35 

velocity  is  4  X  r-^  =  14  f .  s.  above  standard.     From  Table  IV,  the  multiplier  for 

Column  12  is  +.18. 


jV/nc/   =  /^  /tTzoT^^ 


Figure  34. 
Therefore  we  have,  usins:  the  traverse  tables  to  resolve  the  wind  forces: 


Cause  of  error. 

Affects. 

Wind 

ic 

Range 

Deflection.. 
Range 

Atmosphere 

Velocity 

Range 

Range 

Height  of  bull's  eye 

Intentional    deflec- 
tion. 

Range 

Deflection.. 

Formulffi. 


18eos450x4=i^^ 
18  sin  45^x4-=:.^^ 


5X 


39 


10 

.18  X  136 
1.       229 


12  X 


100 
20 


12  X  — 
12 


Errors  on  point  P  as  an  origin  for  standard  conditions. 


Range. 


Short, 
Yds. 


19 


19.5 


Over. 
Yds. 


18.0 


24.5 
64.1 

60.0 


166.6 
19.5 


147.1 
over 


Deflection. 


Right. 
Yds. 


8.5 


8.5 


Left. 

Yds. 


65.0 


65.0 

8.5 


56.5 
left. 


Observed  distance  from  target  in  range 125.0  yds.  short 

Error  (where  shot  should  have  fallen) 147.1  yds.  over 


True  mean  error  in  range  under  standard  conditions 272.1  yds.  short 

Observed  distance  from  line  of  lire  through  bull's  eye  in  deflection.  .  85.0  yds.  left 
Error  (where  shot  should  have  fallen) 56.5  vds.  left 


Tru  e  mean  error  in  deflection  under  standard  conditions 28.5  yds.  left 


223  EXTEEIOE  BALLISTICS 

That  is,  under  standard  conditions,  the  mean  point  of  impact  of  this  gun  is  272,1 
yards  short  of  and  28.5  yards  to  the  left  of  the  point  of  fall  (P)  of  the  perfect  tra- 
jectory of  the  gun  through  the  bull's  eye.  We  want  to  so  adjust  the  sight  scales  as  to 
bring  the  mean  actual  trajectory  of  the  gun  into  coincidence  with  the  perfect  tra- 
jectory of  the  gun;  that  is,  to  shift  the  mean  point  of  impact  of  the  gun  to  its  proper 
theoretical  position,  that  is,  to  the  point  P.    To  do  this  we : 

1.  Run  up  the  sight  in  range  until  the  pointer  indicates  8273.1  yards.  Then 
slide  the  scale  under  the  pointer  until  the  pointer  is  over  8000  yards  on  the  scale. 
Then  clamp  the  scale  in  this  position. 

2.  From  Column  18  of  the  range  table,  we  see  that  28.5  yards  deflection  at  8000 

12 
yards  range  corresponds  to  a  movement  of  28.5  x  ^=5.3  knots  on  the  deflection 

scale.  Therefore  set  the  sight  in  deflection  at  55.3  knots.  Then  slide  the  deflection 
scale  under  the  pointer  until  the  pointer  is  over  50  knots  on  the  scale.  Then  clamp 
the  scale  in  that  position.* 

When  the  above  process  has  been  completed,  the  gun  should  shoot,  under  standard 
conditions,  so  that  the  mean  point  of  impact  will  fall  at  P. 

361.  With  the  sights  adjusted  as  described  above,  under  standard  conditions, 
the  shot  should  fall  at  the  range  and  with  the  deflection  given  by  the  sight  setting ; 
that  is,  the  shot  should  all  fall  at  the  mean  point  of  impact.  And  any  variation  from 
standard  conditions  should  cause  the  errors  indicated  for  such  variations  in  the 
range  tables;  and  such  errors  could  be  easily  handled  by  the  spotter.  Of  course  this 
statement,  if  taken  literally,  means  that  all  errors  have  been  eliminated  from  the  gun, 
and  that  all  shots  fired  from  it  under  the  same  conditions  will  strike  in  the  same  place, 
that  place  being  the  mean  point  of  impact  for  those  conditions.  It  is  of  course  never 
possible  to  actually  accomplish  this,  owing  first  to  the  inherent  errors  of  the  gun, 
and  second  to  unavoidable  inaccuracies  in  the  work.  If  the  work  be  well  done,  how- 
ever, the  result  will  be  to  come  as  near  as  is  humanly  possible  to  that  most  desirable 
perfect  condition. 
Mean  362.  If  the  distance  of  the  point  of  fall  of  each  shot  from  the  mean  point  of 

dispersion.  jj^-,p^(^|.  ]jg  found  for  every  shot  fired,  and  the  arithmetical  mean  of  these  distances  be 
found,  we  have  a  distance  which  is  called  the  "  mean  dispersion  from  the  mean  point 
of  impact."  This  information  is  desirable  because  it  gives  an  idea  of  the  accviracy 
and  of  the  consistency  of  shooting  of  the  gun.  For  example,  one  gun  of  a  battery 
may  have  its  mean  point  of  impact  with  reference  to  a  certain  target  at  a  distance, 
say,  of  100  yards  over  and  25  yards  to  the  right,  but  all  of  its  shot  may  fall  at,  say, 
a  mean  distance  of  only  10  yards  from  the  mean  point  of  impact;  that  is,  its  shot  will 
all  be  well  bunched  and  closely  grouped  around  the  mean  point  of  impact.  Its  mean 
dispersion  from  mean  point  of  impact  is  small,  and  it  is  a  good  gun ;  for  the  spotter 
can  readily  bring  its  shot  on  the  target,  and  when  he  has  done  this  they  will  all  fall 
there.  If,  on  the  contrary,  with  another  gun,  the  mean  point  of  impact  be,  say,  only 
10  yards  over  and  10  yards  to  the  right  of  the  target,  but  the  mean  dispersion  from 
the  mean  point  of  impact  be,  say,  75  yards,  the  shot  will  fall  scattered,  the  spotter 
will  have  difficulty  in  bringing  the  mean  point  of  impact  on  the  target  and  in  keeping 
it  there,  and  after  he  has  done  so  the  percentage  of  hits  will  be  much  smaller  than 

*  For  setting  the  sights  preparatory  to  adjusting  the  scales,  given  the  true  mean 
errors,  we  may  readily  figure  out  the  following  rules: 

„  f  If  the  error  be  "  short,"  add  it  to  the  standard  range. 

"[if  the  error  be  "  over,"  subtract  it  from  the  standard  range. 

P^  „     ..  fif  the  error  be  "  right,"  subtract  its  equivalent  in  knots  from  50. 


■{". 


the  error  be  "  left,"  add  its  equivalent  in  knots  to  50. 


CALIBEATION  OF  SINGLE  GQNS  AXD  A  SHIP'S  BATTEKY     223 


with  the  first  gun.    Comparing  the  two  guns,  we  would  say  that  the  second  gun  was 
a  poor  one  compared  to  the  first. 

363.  All  that  the  spotter  can  hope  to  do,  with  a  single  gun,  is  to  bring  the  mean 
point  of  impact  on  the  target  and  hold  it  there ;  then  if  the  gun  shoots  closely  he  will 
make  the  maximum  possible  number  of  hits.  If,  however,  the  gun  does  not  shoot 
closely,  there  is  nothing  that  can  be  done  to  increase  the  number  of  hits ;  he  is  simply 
doing  the  best  that  he  can  with  an  inferior  weapon.  Similarly,  with  salvo  shooting, 
in  which  the  spotter  tries  to  bring  the  mean  point  of  impact  of  all  the  guns  (that  is, 
the  mean  point  of  impact  of  all  the  mean  points  of  impact  of  all  the  individual  guns) 
on  the  target  (or  slightly  in  front  of  it)  and  keep  it  there.  After  he  has  done  this  the 
result  depends  upon  the  mean  dispersion  of  each  gun  from  its  own  mean  point  of 
impact,  and  upon  the  accuracy  with  which  the  work  of  calibration  has  brought  the 
mean  points  of  impact  of  all  the  guns  into  coincidence  for  the  same  setting  of  the 
sights. 

364.  As  an  example  of  the  mean  dispersion  from  the  mean  point  of  impact,  we 
will  now  determine  that  quantity  for  the  12"  gun  already  calibrated. 


Power  of 
spotter. 


In  range. 

In  deflection. 

No.  of 

shot. 

Fall  relative 

to  target. 

Short  or  over. 

Yds. 

Position  of 
mean  point  of 
impact  rela- 
tive to  target. 
Short  or  over. 
Yds. 

Variation  of 
each  shot 
from  mean 
point  of  im- 
pact.    Short 
or  over. 
Yds. 

Fall  relative 

to  target. 

Right  or  left. 

Yds. 

Position  of 
mean  point  of 

impact  rela- 
tive to  target. 
Right  or  left. 
Yds. 

Variation  of 
each  shot 
from  mean 
point  of  im- 
pact.    Right 
or  left. 
Yds. 

1 

2 

3 

4 

5 

6 

200  short 
150  short 
100  short 
150  short 
100  short 
50  short 

125  short 
125  short 
125  short 
125  short 
125  short 
125  short 

75  short 
25  short 
25  over 
25  short 
25  over 
75  over 

90  left 
95  left 
95  left 
85  left 
75  left 
70  left 

85  left 
85  left 
85  left 
85  left 
85  left 
85  left 

5  left 
10  left 
10  left 

0 

10  right 
15  right 

6|250 
41.7 

yards  in 
range. 

6|50 
8.3 
yards  in  de- 
flection. 

Therefore  the  mean  dispersion  of  this  gun  from  the  mean  point  of  impact  is : 

In  range   42  yards. 

In  deflection 8  yards. 

365.  Note  that  in  finding  the  position  of  the  mean  point  of  impact  we  com-  Mean  point 
bined  the  errors  of  the  several  shots  with  their  proper  algebraic  signs,  because  we  and  mean 
were  finding  the  mathematical  center  of  gravity  of  the  group  of  points  of  fall ;  but 
in  computing  the  mean  dispersion  from  the  mean  point  of  impact  we  were  simply 
trying  to  find  the  average  distance  at  w^hich  the  shot  fell  from  that  point,  without 
regard  to  direction,  and  so  we  discard  the  algebraic  signs  and  simply  take  the 
arithmetical  mean.  It  is  to  be  noted  that  all  corrections  applied  to  the  original  data 
change  all  the  shot  alike,  and  therefore  do  not  change  their  position  relative  to  each 
other.  We  may  therefore  find  the  dispersion  by  using  the  original  data  before  cor- 
rection, as  was  done  above.  Or  we  could  correct  each  shot  separately  and  then  find 
the  dispersion  from  the  results,  but  the  former  process  is  of  course  the  shorter  and 
simpler.  Corrections  to  the  original  data  do  of  course  change  the  positions  of  the  shot 
relative  to  any  fixed  outside  point,  such  as  the  target  or  the  point  P,  and  therefore 
we  have  the  process  previously  employed  for  finding  errors,  etc. 


224 


EXTERIOR  BALLISTICS 
EXAMPLElS. 


1.  For  the  following  results  of  different  calibration  practices,  compute  the  true 
mean  errors  under  standard  conditions  and  the  mean  dispersion  from  mean  point  of 
impact;  and  tell  how  to  adjust  the  sight  scales  in  each  case  in  range  and  deflection 
to  make  the  gun  shoot  as  pointed  when  all  conditions  are  standard. 

14"  gun;  7  =  2600  f.  s.;  w  =  1400  pounds;  c  =  0.70. 


Actual  distance  of  target  from  gun,  yds 

Sights  set  in  range  for,  yds 

Sights  set  in  deflection  for,  l^nots 

Center  of  bull's  eye  above  water,  feet.  . 
Bearing  of  target  from  ship,  °true.  ..  , 

Wind  blowing  from,  °true 

Wind  blowing  with  a  velocity  of,  knots. 

Barometer,  inches 

Temperature  of  air,  °F 

Temperature  of  powder,  °F 

Weight  of  shell,  pounds 

Number  of  shots  fired 

Fall  of— 

Shot  No.  1 

Shot  No.  2 

Shot  No.  3 

Shot  No.  4 


1. 

2. 

3. 

4. 

5. 

6. 

7. 

13000 

13500 

14000 

14500 

13300 

13700 

14200 

13000 

13500 

14000 

14500 

13300 

13700 

14200 

35 

40 

30 

42 

GO 

65 

70 

4 

3 

5, 

6 

4 

5 

6 

45 

180 

80 

315 

270 

250 

345 

180 

225 

0 

90 

180 

250 

165 

12 

15 

18 

20 

25 

15 

18 

28.. 50 

29.00 

29.50 

30.00 

30.50 

31.00 

30.25 

GO 

65 

70 

75 

SO 

85 

90 

80 

85 

95 

100 

97 

82 

75 

1395 

1390 

1405 

1410 

1407 

1393 

1397 

4 

4 

4 

4 

^ 

4 

4 

25 

100 

150 

75 

150 

200 

100 

S. 

s. 

Ov. 

S. 

S. 

Ov. 

s. 

75 

70 

200 

20 

15 

30 

100 

L. 

L. 

L. 

L. 

R. 

R. 

R. 

50 

75 

200 

90 

100 

250 

75 

S. 

S. 

Ov. 

S. 

s. 

Ov 

s. 

100 

75 

150 

10 

20 

35 

80 

L. 

L. 

L. 

L. 

R. 

R. 

R. 

100 

50 

175 

20 

125 

275 

50 

s. 

S. 

Ov. 

Ov. 

S. 

Ov. 

s. 

150 

50 

175 

15 

30 

50 

110 

L. 

L. 

L. 

L. 

R. 

R. 

R. 

75 

25 

130 

10 

110 

225 

0 

S. 

Ov. 

Ov. 

Ov. 

S. 

Ov. 

80 

50 

150 

20 

25 

40 

90 

L. 

L. 

L. 

L. 

R. 

R. 

R. 

14400 
14400 

63 

4 

0 

270 

22 
29.75 

83 

80 

1404 

4 

20 
Ov. 
50 
R. 
25 
Ov. 
55 
R. 
30 
Ov. 
60 
R. 
22 
Ov. 
50 
R. 


ANSWERS. 


1 
2 
3 

4 
5 
6 

7 


True  mean  errors. 


Range. 

Yds. 

11 

7  0v. 

8 

4S. 

41 

2S. 

449 

4S. 

322 

9S. 

479 

2  0v. 

89 

2  0v. 

134 

lOv. 

Deflec- 
tion. 
Yds. 


79. OR. 

78. 5R. 

20.7  R. 
124.0 R. 
148.7  L. 
133.7  L. 

146.7  L. 

168.8  L. 


Mean  dispersion 
from  M.  P.  of  I. 


Range. 
Yds. 


25.0 
37.5 

18.75 
48.75 
16.25 
25.0 
31.25 
3.25 


Deilec- 
tion. 
Yds. 


24.4 
11.3 
18.75 
3.75 
5.00 
6.25 
10.0 
3.75 


Set  sights  for. 


Range. 
Yds. 


12988.3 
13508.4 
14041.2 
14949.4 
13632.9 
13220.8 
14110.8 
14265.9 


Deflec- 
tion. 
Knots. 


42.6 
43.1 
48.5 
40.0 
63.4 
61.6 
62.1 
63.7 


Clamp  scales  at. 


Range. 
Yds. 


13000 
13500 
14000 
14500 
13300 
13700 
14200 
14400 


Deflec- 
tion. 
Knots. 


50 
50 
50 
50 
50 
50 
50 
50 


CALIBEATION  OF  SINGLE  GUNS  AND  A  SHIP'S  BATTERY     325 

2.  For  the  following  results  of  different  calibration  practices,  compute  the  true 
mean  errors  under  standard  conditions  and  the  mean  dispersion  from  mean  point 
of  impact;  and  tell  how  to  adjust  the  sight  scales  in  range  and  deflection  in  each 
case  to  make  the  gun  shoot  as  pointed  under  standard  conditions. 

7"  gun;  7  =  2700  f.  s.;  w  =  16b  pounds;  c=0.61. 


Actual  distance  of  target  from  gun,  yds 

Sights  set  in  range  for,  yds 

Sights  set  in  dellection  at,  knots 

Center  of  bull's  eye  above  water,  feet.  . 
Bearing  of  target  from  ship,  °true.  ..  , 

Wind  blowing  from,  °true 

Wind  blowing  with  velocity  of,  knots.  . 

Barometer,  inches 

Temperature  of  air,  °F 

Temperature  of  powder,  °F 

Weight  of  shell,  pounds 

Number  of  shots  fired 

Fall  of— 

Shot  No.  1 

Shot  No.  2 

Shot  No.  3 

Shot  No.  4 


6000 

6000 

70 

6 

0 

180 
13 
30.50 
50 
95 
170 
4 

50 
Ov. 
150 

R. 

75 
Ov. 
165 

R. 

70 
Ov. 
155 

R. 

65 
Ov. 
150 

R. 


6200 

6200 

65 

5 

180 
180 
17 
30.25 
55 
97 
172 
4 

75 
Ov. 
100 
R. 
50 
Ov. 
95 
R. 
25 
Ov. 
70 
R. 
25 
S. 
75 
R. 


6500 

6500 

63 

4 

90 
180 
12 
30.00 
60 
100 
168 
4 

75 

S. 
95 
R. 

55 
S. 
90 
R. 
70 
S. 
85 
R. 
75 
S. 
90 
R. 


6700 
6700 

60 
3 

180 

90 

20 
29.33 

65 

98 

162 
4 

200 
Ov. 

75 

R. 
250 
Ov. 

50 

R. 
200 
Ov. 

25 

R. 
175 
Ov. 

10 

L. 


5. 


7000 

7000 

55 

4 

350 

220 

25 

29.67 

70 

93 

160 

4 

150 

S. 

5 

R. 
125 

S. 

10 

R. 
100 

s. 

0 
130 

s. 

5 

L. 


7200 

7200 

45 

5 

220 

350 

18 

29.00 

75 

87 

161 

4 

100 
Ov. 

5 

R. 
110 
Ov. 

10 

L. 
125 
Ov. 

25 

L. 
117 
Ov. 

20 

L. 


7500 

7500 

43 

6 

160 
30 
30 


7300 
7300 

40 

5 
225 
270 

22 


29.10  28.90 
80    85 


85 

164 

4 

250 
Ov. 
20 
L. 
275 
Ov. 
25 
L. 
300 
Ov. 
30 
L. 
280 
Ov. 
27 
L. 


80 

169 

4 

100 

s. 

25 
L. 

125 
S. 
30 
L. 
130 
S. 
35 
L. 
135 
S. 
40 
L. 


ANSWERS. 


True  mean  errors. 

Mean  Dispersion 
from  M.  P.  of  I. 

Set  sights  at. 

Clamp  scales  at. 

Range. 

Yds. 

Deflec- 
tion. 

Yds. 

Range. 
Yds. 

Deflec- 
tion. 

Yds. 

Range. 

Yds. 

Deflec- 
tion. 
Knots. 

Range. 

Yds. 

Deflec- 
tion. 
Knots. 

1 

91.8  0V. 
88.7  0V. 

141.7  S. 

19.2  0  V. 
312.5  S. 

39.9  S. 
142.10V. 

63.7  S. 

63.3  R. 

13.7  R. 
40.9  R. 
39. 2L. 

47.8  L. 
33.6  R. 
13.6  L. 
45.3  R. 

7.5 
31.25 

6.85 
21.9 
13.75 

8.0 
16.25 
11.25 

5.0 
12.5 

2.5 
27.5 

6.25 
10.0 

3.0 

5.0 

5908.2 
6111.3 
6641.7 
6680.8 
7312.5 
7239.9 
7357.9 
7363.7 

36.2 
47.1 
42.0 
57.5 

58.6 
44.2 
52.2 
42.2 

6000 
6200 
6500 
6700 
7000 
7200 
7500 
7300 

50 

2 

50 

3 .      . 

50 

4 

50 

5 

50 

6 

50 

7 

50 

8 

50 

15 


CHAPTEE  19. 


Reasons  for 

calibrating 

battery. 


Standard  gun. 


THE  CALIBRATION  OF  A  SHIP'S  BATTERY. 

366.  In  the  preceding  chapter  we  have  seen  how  a  single  gun  is  calibrated  and 
the  sights  so  adjusted  that,  so  far  as  the  inherent  errors  of  the  gun,  etc.,  will  permit, 
the  gun  will  shoot,  under  standard  conditions,  as  the  sights  indicate.  It  was  stated 
that  it  is  not  possible  to  accomplish  this  result  with  absolute  accuracy.  If  it  were, 
we  could  adjust  the  sights  of  each  gun  of  the  battery  separately,  and  then,  if  they 
were  all  mechanically  just  alike,  we  would  have  all  the  shot  from  each  gun  falling  at 
its  mean  point  of  impact  (within  the  limits  of  inherent  errors),  and  the  mean  points 
of  impact  of  all  the  guns  would  be  the  same.  As  a  matter  of  fact,  however,  the  mean 
points  of  impact  of  the  several  guns  would  not  coincide,  if  this  method  were  followed, 
and  of  course  all  the  shot  from  any  one  gun  would  not  all  fall  at  its  mean  point  of 
impact.  Some  remarks  were  made  in  the  last  chapter  relative  to  the  necessity  for 
getting  the  guns  so  calibrated  that  the  shot  from  all  of  them  Avill  fall  together  for  the 
same  sight  setting,  and,  as  a  matter  of  fact,  this  is  more  important  than  it  is  to  get 
them  so  that  actual  and  sight-bar  ranges  coincide  under  standard  conditions.  Con- 
ditions are  almost  never  standard  during  firing,  and  even  if  they  were  there  are  many 
other  factors  which  prevent  the  actual  and  the  sight-bar  ranges  from  being  the  same. 
But  if  the  mean  points  of  impact  of  the  several  guns  for  the  same  sight  setting  can  be 
brought  very  nearly  into  coincidence,  then  any  variation  of  the  resultant  point  from 
the  target  (that  is  of  difference  between  actual  and  sight-bar  ranges)  can  be  readily 
handled  by  the  spotter.  This  means  that  if  the  salvos  are  well  bunched  the  spotter 
can  control  the  fire  successfully,  but  if  the  shots  are  scattered  he  cannot.  We  will 
now  proceed  with  an  entire  battery  to  bring  all  guns  to  shoot  together. 

367.  Having  calibrated  each  gun  separately,  as  described  in  the  preceding 
chapter,  we  now  proceed  to  select  a  gun  as  the  "  standard  gun,"  to  the  shooting  of 
which  we  propose  to  make  that  of  all  the  others  conform,  providing  the  performance 
of  any  one  gun  is  good  enough  to  justify  selecting  it  for  the  purpose.  From  what  we 
have  seen  in  the  preceding  chapter  we  would  naturally  select  one  whose  mean  dis- 
persion from  mean  point  of  impact  is  small,  that  is,  one  that  bunches  its  shots ;  and, 
other  things  being  equal,  if  we  have  one  whose  sights  are  very  nearly  in  adjustment, 
we  will  use  that  one  without  changing  the  sight  adjustment.  Any  gun  may  of  course 
be  selected  as  the  standard,  and  the  sights  of  the  others  brought  to  correspond  to  it, 
but  the  considerations  set  forth  above  would  naturally  govern,  as  a  matter  of  common 
sense.  If  no  gun  be  sufficiently  accurate,  or  if  none  has  its  sights  sufficiently  well 
adjusted  to  justify  its  selection  as  a  standard  gun,  then  we  must  correct  all  guns  to  the 
mean  point  of  impact.  The  practical  method  of  bringing  the  sights  of  a  number  of 
guzis  to  correspond  is  best  shown  by  an  actual  problem. 


CALIBRATION  OF  SINGLE  GUNS  AND  A  SHIP'S  BATTERY     227 


368.  The  results  of  the  individual  calihration  of  a  battery  of  eight  13"  guns    Calibration 
(y  =  2900  f.  s.,  ft)  =  870  pounds,  c=0.(.)l),  at  an  actual  range  of  8000  yards,  were 
as  follows : 


True  mean  errors  under  standard  conditions. 
Yards. 

Number  of  gun. 

In  range. 

In  deflection. 

Short. 
Yards. 

Over. 
Yards. 

Right. 
Yards. 

Left. 
Yards. 

1 

100 
75 

"so 

25 
5 

"40 
90 

'125 

'26 
15 
10 

5 

15 

9 

3 

4 

15 

6 

20 

7 

8 

20 

It  is  desired  to  calibrate  the  above  battery.  From  an  examination  of  the  above 
results,  assuming  that  the  eight  guns  are  equally  good  in  the  absence  of  any  knowledge 
to  the  contrary,  we  will  select  No.  7  as  the  standard  gun;  and,  as  its  sights  are  very 
slightly  out,  we  will  leave  them  unchanged  and  bring  the  sights  of  the  other  guns  to 
correspond  with  them. 

The  work,  which  is  best  expressed  in  tabular  form,  then  becomes  (all  guns  were 
fired  with  sights  set  at  8000  yards  in  range  and  50  knots  in  deflection)  : 


With  reference  to  standard  gun,  each 
gun  shot. 

To  bring  all  sights  together  set 
them  for  each  gun  as  follows: 

Number  of 
gun. 

In  range. 
Yds. 

In  deflection. 

In  range. 
Yds. 

In  deflection. 

Yards. 

Knots.* 

Knots. 

1 

95  short 
70  short 
45  over 
95  over 
75  short 
20  short 
Standard 
130  over 

20  left 
15  right 
10  right 
5  right 
20  left 
25  left 
Standard 
25  left 

3.7  left 

2.8  right 

1.9  right 
1.0  right 
3.7  left 
4.6  left 
Standard 
4.6  left 

8095 
8070 
7955 
7905 
8075 
8020 
Standard 
7870 

53  7 

2 

47  2 

3 

48  1 

4 

49  0 

53  7 

6 

54  6 

7 

8 

54  6 

12 
*  From  the  range  table,  at  8000  yards,  one  yard  in  deflection  corresponds  to  7^^..  knots 

on  the  deflection  scale. 


After  the  sights  have  been  set  as  indicated  in  the  two  right-hand  columns  of  the 
above  table,  move  the  sight  scales  under  the  pointers  until  the  pointers  are  over  the 
8000-yard  mark  in  range  and  the  50-knot  mark  in  deflection  in  each  case,  and  then 
clamp  the  scales  in  those  positions.  The  guns  are  then  calibrated  to  shoot  together. 
It  will  be  noted  that,  theoretically,  we  should  have  set  the  range  scales  for  the  5  yards 


228  EXTEPtlOE  BALLISTICS 

short  in  range  and  the  deflection  scales  for  the  5  yards  right  in  deflection  of  the 
standard  gun,  to  be  absolutely  accurate ;  but,  as  the  sight  scales  are  graduated  to 
50-yard  increments  in  range  only,  it  is  impracticable  to  go  any  closer  in  range.  It 
would  perhaps  be  well  to  adjust  each  deflection  scale  to  51  knots  instead  of  50,  in 
order  to  allow  for  the  5  yards  right  deflection  of  the  standard  gun. 
Different  369.  When  we  wish  to  calibrate  a  ship's  battery  that  is  composed  of  separate 

batteries  of  different  calibers,  we  calibrate  each  caliber  by  itself,  as  already  described. 
The  difference  between  the  mean  points  of  impact  of  the  standard  guns  of  the  differ- 
ent calibers  will  be  the  difference  between  the  centers  of  impact  of  the  salvos  from  the 
several  calibers,  and  this  must  be  allowed  for  in  firing  all  calibers  together.  To 
attempt  to  calibrate  the  sights  of  all  calibers  together  by  a  readjustment  of  the  sight 
scales  would  not  be  wise ;  for  if  they  could  be  brought  to  shoot  together  at  one  range 
in  this  way,  it  would  necessarily  ensure  dispersion  of  the  several  salvos  at  all  other 
ranges.  Therefore  the  only  practical  way  of  handling  this  proposition  is  to  deter- 
mine the  error  of  each  caliber  at  the  range  in  use  and  apply  it  properly  in  sending 
the  ranges  to  the  guns;  which  means  send  different  ranges  to  the  guns  of  different 
calibers,  so  related  that  the  results  will  bring  the  mean  points  of  impact  of  the 
several  calibers  together  at  the  range  in  use.  As  far  as  possible,  these  differences  in 
ranges  should  be  tabulated  for  different  ranges.  As  the  fire-control  system  is 
arranged,  as  a  rule,  to  permit  the  control  of  each  caliber  battery  independently  of 
the  others,  this  method  presents  no  difficulties  other  than  a  little  care  on  the  part  of 
the  spotter  group. 

370.  For  instance,  suppose  that  we  have  a  ship  with  a  mixed  battery  of  7",  8"  and 
12"  guns;  that  each  of  these  calibers  has  been  calibrated  at  8000  yards;  and  has  had 
the  mean  point  of  impact  of  its  salvos  located  with  reference  to  the  target  as  follows : 

7"  battery 100  yards  over 3  knots  right. 

8"  battery 50  yards  short 3  knots  left. 

12"  battery 150  yards  over 2  knots  left. 

Then  if  we  wish  to  fire  broadside  salvos  from  this  entire  battery,  the  ranges  and 
deflections  should  be  sent  to  the  guns  as  follows,  for  8000  yards : 

To  the     7" 7900  yards 47  knots  deflection. 

To  the     8" 8050  yards 53  knots. 

To  the  12" 7850  yards 52  knots. 


CALIBEATION  OF  SINGLE  GUNS  AND  A  SHIP'S  BATTERY     229 

EXAMPLES. 

1.  Having  determined  the  true  mean  errors  of  guns  under  standard  conditions, 
by  calibration  practice,  to  be  as  given  in  the  following  table;  how  should  the  sights 
of  each  caliber  be  adjusted  to  make  all  the  guns  of  that  caliber  shoot  together?  (Six 
separate  problems.) 


True 

mean  errors  of 

guns  under  standard 

conditions. 

6"— 

G. 

7"— J. 

8"— K. 

12"— N, 

13"— P. 

14"—  R. 

d 

Errors  at 

Errors  at 

Errors  at 

Errors  at 

Errors  at 

Errors  at 

b£ 

range  o 

f  4500 

range  of  6500 

range  of  8500 

range  of  10000 

range  of  11000 

range  of  13000 

yar 

Is. 

yards. 

yards. 

yards. 

yards. 

yards. 

S 

Eange. 

Defl. 

Range. 

Defl. 

Range. 

Defl. 

Range. 

Defl. 

Range. 

Defl. 

Range. 

Defl. 

^ 

Yds. 

Yds. 

Yds. 

Yds. 

Yds. 

Yds. 

Yds. 

Yds. 

Yds. 

YMs. 

Yds. 

Yds. 

1.. 

25 

20 

50 

30 

100 

5 

100 

15 

125 

25 

75 

15 

S. 

R. 

Ov. 

L. 

R. 

R. 

Ov. 

L. 

Ov. 

L. 

R. 

L. 

2. . 

50 

30 

75 

40 

120 

15 

75 

10 

100 

30 

100 

15 

Ov. 

R. 

Ov. 

L. 

R. 

L. 

Ov. 

L. 

Ov. 

L. 

Ov. 

R. 

3.. 

75 

35 

100 

25 

90 

20 

50 

20 

75 

25 

75 

2(1 

Ov. 

L. 

Ov. 

R. 

Ov. 

L. 

Ov. 

R. 

Ov. 

R. 

Ov. 

R. 

4.. 

5 

5 

75 

15 

75 

25 

0 

5 

100 

30 

50 

20 

R. 

L. 

S. 

L. 

Ov. 

R. 

R. 

S. 

R. 

8. 

L. 

5.. 

30 

30 

125 

10 

100 

20 

100 

25 

100 

25 

Ov. 

R. 

S. 

L. 

Ov. 

R. 

R. 

R. 

s. 

L. 

6.. 

50 

25 

10 

70 

SO 

10 

90 

15 

100 

30 

S. 

L. 

s. 

R. 

R. 

L. 

R. 

R. 

Ov. 

R. 

7.. 

100 

40 

100 

25 

70 

25 

75 

30 

>  •  •  • 

to 

20 

s. 

R. 

Ov. 

R. 

R. 

R. 

Ov. 

L. 

Ov. 

R. 

8.. 

100 

40 

90 

20 

70 

30 

100 

30 

•  •  •  • 

70 

15 

Ov. 

L. 

S. 

L. 

Ov. 

L. 

R. 

R. 

S. 

L. 

AN8WERS. 


To  bring  all  sights  together  for  each  caliber,  set  the  sights  for  that  caliber  as  given 
below,  and  then  slide  scales  to  standard  readings  and  clamp. 


be 

6"- 

-G. 

7"— 

J. 

8"— 

K. 

12"- 

N. 

13"— 

P. 

14"— 

R. 

a 

3 

Range. 

Defl. 

Range. 

Defl. 

Range. 

Defl. 

Range. 

Defl. 

Range. 

Defl. 

Range. 

Defl. 

Yds. 

Yds. 

Yds. 

Kts. 

Yds. 

Kts. 

Yds. 

Kts. 

Yds. 

Kts. 

Yds. 

Kts. 

1.. 

4520 

43.0 

6450 

56.0 

8600 

49.25 

9900 

52.8 

10875 

52.0 

13075 

51.35 

2.. 

4445 

40.2 

6425 

58.0 

8620 

52.25 

9925 

52.1 

10900 

.52.4 

12900 

48.65 

3.. 

4420 

58.4 

6400 

45.0 

8410 

53.00 

9950 

47.9 

10925 

48.0 

12925 

48.20 

4.. 

Rtan 

dard. 

6575 

53.0 

8425 

46.25 

Rtan 

dard. 

11100 

47.6 

13050 

51.80 

5.  . 

4465 

40.2 

6625 

52.0 

8400 

47.00 

10100 

47.2 

13100 

52.25 

(5.  . 

4545 

55.6 

6510 

36.0 

8580 

51.50 

10090 

48.6 

12900 

47.30 

7.  - 

4595 

37.4 

6400 

45.0 

8570 

46.25 

9925 

54.9 

12925 

48.20 

8.. 

4395 

59.8 

6590 

54.0 

8430 

54.50 

10100 

46.5 

13070 

51.35 

PAET  VI. 

THE  ACCURACY  AND  PROBABILITY  OF  GUN  FIRE 
AND  THE  MEAN  ERRORS  OF  GUNS. 

INTEODUCTIOX  TO  PAET  VI. 

"We  have  now  learned  all  that  mathematical  theory  can  teach  us  with  certainty 
about  the  flight  of  a  projectile  in  air,  about  the  errors  that  may  be  introduced  into 
such  flight  by  known  causes,  and  about  the  methods  of  compensating  for  such  errors. 
After  all  this  has  been  done  there  must,  in  the  nature  of  things,  remain  certain 
errors  that  cannot  be  either  eliminated  or  covered  by  strict  mathematical  theories, 
and  such  errors  are  known  as  the  inherent  errors  of  the  gun.  It  is  the  purpose  of 
Part  VI  to  discuss  the  general  nature  of  these  errors,  their  methods  of  manifesting 
themselves,  and  their  probable  effect  upon  the  accuracy  of  fire. 


CHAPTER  20. 


THE  ERRORS  OF  GUNS  AND  THE  MEAN  POINT  OF  IMPACT.  THE  EaUATION 
OF  PROBABILITY  AS  APPLIED  TO  GUN  FIRE  WHEN  THE  MEAN  POINT 
OF  IMPACT  IS  AT  THE  CENTER  OF  THE  TARGET. 


Z. 

Y. 


(zi,  !/i,  etc.) . 

%■ 

n. 

yx- 
Jy 
yz. 


New  Symbols  Introduced. 

.  Axis  of;  axis  of  coordinates  lymg  along  range,  for  points  over  or 

short  of  the  target. 
.  Axis  of;  axis  of  coordinates  in  vertical  plane  through  target,  for 

points  above  or  below  the  center  of  the  target. 
.  Axis  of ;  axis  of  coordinates  in  vertical  plane  through  target,  for 

points  to  right  or  left  of  center  of  the  target. 
.  Coordinates  of  points  of  impact  in  vertical  plane  through  target. 
.  Summation  of  z^,  Zo,  etc. 
.  Summation  of  y^,  t/„,  etc. 
.  Number  of  shots. 
.  ]\Iean  error  in  range. 
.  Mean  vertical  error. 
.  Mean  lateral  error. 


371.  Before  proceeding  to  the  discussion  of  the  accuracy  and  probability  of  gun 
fire,  it  is  wise  to  collect  and  consider  certain  definitions  and  descriptions  of  which  a 
full  understanding  is  necessary  in  order  to  clearly  understand  what  is  to  follow. 

372.  There  are  three  general  classes  of  errors  which  enter  into  gun  fire,  and  the   classes  of 
distinction  between  which  must  be  clearly  comprehended.     They  may  be  stated  as   ^"°^^' 
follows : 

1.  Errors  Resulting-  from  Mistakes  or  Accidents. — As  examples  of  these  may  be  Mistakes, 
mentioned  mistakes  in  estimating  ranges  or  deliections,  mistakes  in  sight  setting, 
mistakes  in  pointing,  etc.  These  are  matters  that  pertain  to  the  training  of  the 
personnel,  and  of  course  have  no  place  in  any  discussion  of  the  principles  of  ballistics, 
etc.,  for  no  theory  can  be  developed  unless  all  such  causes  of  error  are  first  eliminated. 
Such  mistakes  of  course  cause  poor  shooting,  but  they  have  no  place  in  any  theoretical 
investigation  of  the  performance  of  the  gun. 

3.  Preventable  Errors. — These  are  errors  arising  from  causes  which  must  Preventable, 
necessarily  exist,  but  in  regard  to  which  the  theories  are  well  understood,  and  for 
which  it  is  possible  to  compensate  by  a  practical  application  of  such  theories.  Ex- 
amples of  this  class  are  the  errors  due  to  wind,  to  variation  in  the  temperature  of 
the  powder,  etc.  One  of  the  principal  provinces  of  the  science  of  exterior  ballistics 
is  to  teach  the  principles  governing  such  errors,  and  to  show  how  they  may  be  over- 
come. In  general,  it  may  be  said  that  it  is  the  principal  duty  of  the  spotter  to  discover 
the  magnitude  and  direction  of  these  errors  and  to  give  the  instructions  necessary 
to  compensate  for  them. 

3.  Unpreventable  Errors. — These  may  be  generally  classified  as  the  inherent   unpre^ 
errors  of  the  gun.    That  is,  they  are  the  result  of  the  very  many  elements  entering 
into  the  shooting  which  cause  variations  in  successive  shots  even  when  most  carefully 
fired  under  as  nearly  as  possible  the  same  physical  conditions,  and  which  therefore 
ensure  that  any  considerable  number  of  shots  from  the  same  gun  will  have  their 


ventable. 


234 


EXTERIOR  BALLISTICS 


Summary. 


Mean  point 
of  impact. 


Mean 
trajectory. 


Deviation  or 
deflection, 


successive  points  of  impact  more  or  less  scattered  about  within  a  certain  area.  These 
causes  are  probably  very  numerous,  it  not  even  being  certain  that  we  have  yet  been 
able  to  recognize  them  all,  and  no  satisfactory  laws  governing  them  have  as  yet  been 
discovered,  nor  is  it  probable  that  such  laws  ever  will  be  determined. 

373.  To  summarize,  it  may  be  said  that  before  entering  on  any  theoretical 
investigation  of  the  subject  of  gunnery  we  must  first  throw  out  all  errors  resulting 
from  mistakes.  We  may  then,  by  the  study  of  exterior  ballistics,  learn  certain 
principles  governing  the  errors  produced  by  certain  known  causes,  and  in  conse- 
quence may  learn  how  to  eliminate  such  errors  from  our  shooting.  When  all  this 
has  been  done,  however,  we  necessarily  have  left  certain  other  causes  of  error  which, 
although  not  great  as  compared  with  the  others,  are  still  sufficient  to  cause  a 
scattering  of  the  points  of  impact  of  successive  shots  from  the  same  gun,  even  when 
fired  under  similar  physical  conditions.  We  manifestly  cannot  hope  to  eliminate 
these  inherent  errors,  and  therefore  we  must  accept  them  as  they  are ;  all  that  we  can 
do  in  regard  to  them  is  to  investigate  their  probable  effect  upon  the  results  of  our 
shooting.  It  is  this  investigation  that  is  to  be  undertaken  in  the  last  two  chapters  of 
this  book,  and  it  is  to  be  noted  that  here  we  cannot  speak  of  anything  as  a  certainty, 
even  in  a  theoretical  and  mathematical  way,  but  can  only  say  that,  mathematically,  it 
is  probable  or  improbable  that  a  certain  thing  will  happen,  and  in  addition  attempt  to 
measure  the  degree  of  probability  or  improbability  which  attaches  to  a  certain  effort. 

374.  Mean  Point  of  Impact. — Let  us  suppose  that  all  errors  except  the  inherent 
errors  of  the  gun  have  been  eliminated,  and  that  a  large  number  of  shots  be  fired, 
under  as  nearly  the  same  physical  conditions  as  possible,  at  a  vertical  target  screen  of 
sufficient  size  to  receive  all  the  shot  under  such  conditions.  Manifestly,  if  there  were 
no  errors  of  any  kind  whatsoever,  all  these  shot  would  describe  the  same  trajectory 
and  strike  the  target  at  the  same  point.  Of  course  this  result  can  never  be  attained 
in  practice,  and  the  many  causes  of  inherent  error  tend  to  scatter  the  several  shot 
about  the  target,  and  only  a  certain  percentage  of  absolute  efficiency  can  be  secured, 
no  matter  how  skillfully  the  gun  may  be  handled.  The  point  which  is  at  the 
geometrical  center  of  all  the  points  of  impact  on  the  screen  is  known  as  the  "  mean 
point  of  impact,"  and  is  of  course  the  center  of  gravity  of  the  'group  of  points  of 
impact.  We  may  also  speak  of  the  mean  point  of  impact  in  the  horizontal  plane  as 
well  as  in  the  vertical  plane  as  given  above. 

375.  Mean  Trajectory. — The  mean  trajectory  of  the  gun  for  these  conditions 
is  the  trajectory  from  the  gun  to  the  mean  point  of  impact.  It  is  manifestly  the 
trajectory  over  which  all  the  shot  would  travel  were  there  no  errors  of  any  kind 
whatsoever. 

376.  Deviation  or  Deflection. — Suppose  Figure  35  to  represent  the  vertical 
target  screen,  the  point  0  at  the  center  being  the  point  aimed  at.  Suppose  the  shot 
struck  at  the  point  P.  Then  the  deviation  or  deflection  of  the  shot  from  the  point 
aimed  at  is  the  distance  OP  in  the  direction  shown.  So  considered,  however,  for 
manifest  reasons,  this  information  is  not  useful,  so  it  is  usual  to  speak  of  the  hori- 
zontal deviation  or  deflection,  which  is  a,  and  of  the  vertical  deviation  or  deflection 
of  the  shot,  which  is  h.  And  algebraic  signs  are  assigned  to  these  deviations  or  deflec- 
tions, -t-  being  above  or  to  the  right  and  —  below  or  to  the  left.  Thus  the  deviations 
or  deflections  of  the  four  points  of  impact  shown  in  Figure  35  would  be: 

For  P    +a      and   +&.         For  P" -a"    and   -&". 

For  P'   ....-a'     and   +&'.        For  P'" .  . .  . -|- a'"  and   -h'". 


In  place  of  the  signs  we  might  speak  of  horizontal  deviations  as  being  to  the  right  or 
to  the  left,  and  of  vertical  deviations  as  being  above  or  below.     In  addition  to  the 


ACCUEACY  AND  PROBABILITY  OF  GUN  FIRE  235 

above  we  may  consider  the  deviation  (the  term  "deflection"  i»  not  ordinarily  used 
in  this  connection)  in  range,  as  the  shot  falls  short  of  or  beyond  the  target.  These 
are  denoted  by  the  +  sign  for  a  shot  that  goes  beyond  and  by  a  —  sign  for  one  that 
falls  short,  but  usually  the  words  "  short "  and  "  over "  are  used  instead  of  the 
algebraic  signs,  and  such  shots  are  spoken  of  as  "  shorts  "  or  "  overs,"  as  the  case 
may  be. 


-  ti  — 

—  —A 

1                       ^-\ 

y 

1 
1 

1                    / 

\. 

\-6- 

A"             / 
'D    1        / 

1    / 

~7^'~'' 

St 

•p"  -^" 

Figure  35. 


377.  Deviation  or  Deflection  from  Mean  Point  of  Impact. — In  the  preceding  Deviation 
paragraph  Ave  explained  deviation  or  deflection  from  any  given  point  of  aim;  the   po^nt  of 
results  giving  the  actual  amount  by  which  the  shot  missed  the  point  aimed  at.     In   '™^*'^ ' 
theoretical  consideration  of  the  accuracy  of  a  gun,  however,  it  is  customary  to  assume 

that  the  point  of  reference  or  origin  of  coordinates  is  the  mean  point  of  impact, 
rather  than  any  given  point  of  aim,  and  our  results  are  then  the  "  deviations  or 
deflections  from  the  mean  point  of  impact."  As  the  mean  point  of  impact  is,  by 
definition,  the  center  of  gravity  of  the  group  of  impacts  caused  by  a  large  number  of 
shot,  it  is  evident  that  the  summation  of  all  the  deviations  from  the  mean  point  of 
impact  must  be  equal  to  zero. 

378.  Dispersion. — Now  suppose  that,  in   Figure  35,  we  had  disregarded  the   Dispersion, 
algebraic  signs,  and  considered  only  the  actual  distances  of  the  points  of  impact 

from  the  point  aimed  at.  The  distance  OF  in  this  case  would  be  the  "  dispersion  '* 
for  the  single  shot;  but  again  it  is  customary  to  separate  the  distance  in  the  two 
directions,  and  we  would  have  a  "  horizontal  dispersion  "  of  a  and  a  "  vertical  dis- 
persion "  of  h ;  although  it  is  not  customary  or  appropriate  to  speak  of  the  "dis- 
persion "  of  a  single  shot,  the  word  being  collective  in  its  nature.  The  "  mean  dis- 
persion "  of  the  four  shots  shown  in  Figure  35  from  the  point  aimed  at  would  be: 

Mean  lateral  dispersion  — —  ' 

Mean  vertical  dispersion     "^  ^^ 

4 

We  may  also  have  "  dispersion  in  range  "  as  well  as  in  the  vertical  plane. 


236 


EXTERIOR  BALLISTICS 


Mean  disper-  379.  Mean  Dispcrsion  from  Mean  Point  of  Impact. — Suppose  we  ajrain  consider 

sion  from  .  .  .  „  .  .     .  .      .  *" 

mean  point  our  clispersions  from  the  mean  point  of  impact  as  an  origin.  Tiien  it  is  evident  that 
the  "  mean  dispersion  from  mean  point  of  impact,"  or  "  mean  dispersion  "  as  it  is 
usually  called,  is  the  average  distance  or  arithmetical  mean  of  the  distances  of  the 
points  of  impact  of  all  the  shot  from  the  mean  point  of  impact.  Now  as  this  latter 
point  is  the  one  at  which  every  theoretically  perfect  shot  should  strike,  it  is  evident 
that  the  mean  dispersion  from  mean  point  of  impact  gives  us  a  measure  of  the 
accuracy  of  the  gun,  that  is,  of  the  extent  to  which  its  shooting  is  affected  by  its 
inherent  errors. 

380.  It  must  be  said  in  regard  to  the  above  definitions,  that  the  terms  defined 
are  often  very  loosely  and  more  or  less  interchangeably  used  in  service.  The  term 
"  deflection  "  is  ordinarily  used  only  to  represent  lateral  displacement,  in  either  the 
vertical  or  the  horizontal  planes ;  and  the  term  "  deviation  "  is  used  for  either  vertical 
or  lateral  displacement  or  for  displacement  in  range,  in  which  case  the  terms  "  vertical 
deviation,"  "  lateral  deviation  "  or  "  deviation  in  range  "  are  customarily  used.  There 
is  also  confusion  in  the  use  of  the  terms  as  to  whether  deviation  or  deflection  from  the 
point  aimed  at  or  from  the  mean  point  of  impact  is  meant.  The  term  "  dispersion  " 
is  fairly  regularly  used  as  defined  above,  but  even  here  the  point  as  to  whether  dis- 
persion from  the  point  aimed  at  or  from  the  mean  point  of  impact  is  meant  is  often 
left  obscure.  The  context  of  the  conversation  or  written  matter  will  usually  show 
what  is  meant.    In  this  book  the  terms  will  be  used  strictly  as  defined. 


Figure  36 


CrUTL 


System  of 
coordinates. 


381.  For  use  in  these  last  two  chapters  we  will  also  introduce  a  special  system  of 
coordinates,  as  shown  in  Figure  36.  This  figure  represents  a  perspective  view  of  a 
vertical  target  screen  of  which  0  at  the  center  is  the  mean  point  of  impact  in  the 
vertical  plane.  The  axis  of  X  is  the  line  from  the  muzzle  of  the  gun  to  0,  the  mean 
trajectory  being  shown.  Z  is  the  horizontal  axis  and  Y  the  vertical  axis  through  the 
center  of  the  target.  It  will  be  noted  that  in  this  system  the  axes  of  X  and  Z  are 
interchanged  from  what  they  ordinarily  are  in  geometry  of  three  dimensions;  and 
this  is  done  in  this  particular  subject  to  preserve  the  convention  that  has  been  con- 
sistently used  throughout,  that  X  and  all  functions  thereof  represent  quantities 
pertaining  to  the  range.    In  this  system  of  coordinates  it  will  be  seen  that,  the  mean 


ACCURACY  AND  PEOBABILITY  OP  GUN  FIHE  237 

point  of  impact  at  the  center  of  the  screen  being  considered  as  the  origin,  coordinates 
{z,  y)  will  definitely  locate  any  point  of  impact  on  the  screen,  while  coordinates 
{x,  z)  will  definitely  locate  any  point  of  impact  on  the  horizontal  plane  through  0. 
And  here  it  may  be  stated  that  it  rarely  becomes  necessary  to  consider  hits  in  the 
vertical  and  in  the  horizontal  plane  together.  Therefore  for  hits  in  the  vertical 
plane  we  use  as  an  origin  the  mean  point  of  impact  in  the  vertical  plane,  and  for  hits 
in  the  horizontal  plane  we  use  as  an  origin  the  mean  point  of  impact  on  the  surface 
of  the  water.  In  Figure  3G,  0  being  the  mean  point  of  impact  in  the  vertical  target 
screen,  the  mean  point  of  impact  in  the  horizontal  plane  of  the  water  would  lie 
behind  the  target  at  the  point  where  the  mean  trajectory  through  0  strikes  the  surface 
of  the  water. 

382.  Having  cleared  up  these  preliminary  matters,  and  bearing  in  mind  that 
all  errors  have  been  eliminated  except  the  inherent  errors  of  the  gun,  it  may  now  be 
stated  that  it  becomes  important,  under  these  conditions,  to  be  able  to  answer  certain 
questions  in  regard  to  the  probability  of  securing  hits  under  given  conditions.  For 
instance,  with  a  properly  directed  fire  from  which  all  avoidable  errors  have  been 
removed,  what  are  the  chances  of  hitting  a  given  target  at  a  given  range;  what 
proportion  of  the  total  number  of  shot  fired  at  it  may  reasonably  be  expected  to  hit 
it,  etc.? 

383.  In  other  words,  the  preceding  chapters  having  taught  us  the  methods  to  be 
followed  in  eliminating  all  possible  sources  of  error  or  of  compensating  for  their 
effects,  we  now  wish  to  conduct  an  investigation  that  will  enable  us  to  determine  what 
are  our  chances  of  hitting  under  given  conditions.  From  the  results  of  this  investiga- 
tion, applied  to  any  particular  case,  we  can  tell  how  much  of  a  drain  it  would  probably 
be  upon  our  total  ammunition  supply  to  make  an  effective  attack  under  given  con- 
ditions, and  hence  whether  or  not  we  can  afford  to  make  the  attempt.  To  arrive  at 
answers  to  such  questions  we  must  fall  back  upon  the  theory  of  probability. 

384.  It  will  be  readily  understood  from  what  has  been  said  that  the  deviations   Deviations 
of  projectiles  from  their  mean  point  of  impact  are  closely  analogous  to  what  are   dental 
called  "  accidental  errors  "  in  the  text  books  on  the  subjects  of  probability  and  least 
squares;  such,  for  example,  as  errors  that  are  made  in  the  direct  measurement 

of  a  magnitude  of  any  kind ;  and  they  obey  the  same  laws.  Small  deviations  are  more 
frequent  than  large  ones;  positive  and  negative  deviations  are  equally  probable  and 
therefore  equally  frequent,  if  the  number  of  shot  be  great ;  very  large  deviations  are 
not  to  be  expected  at  all  (if  one  occur  it  must  be  the  result  of  some  mistake  or  some 
avoidable  error). 


238 


EXTERIOE  BALLISTICS 


385.  Suppose  that  we  have  as  the  point  of  aim  the  center,  0,  of  the  vertical 
target  screen  shown  in  Figure  37,  and  suppose  we  had  n  points  of  impact  as  shown 
(18  are  shown)  of  which  the  coordinates  are  {z^,  yj,  (z^,  y^),  {zn,  yn),  each  with  its 
proper  algebraic  sign.    Then  manifestly  the  coordinates  of  the  mean  point  of  impact 

referred  to  0  as  an  origin  are  (^^  ,  -^  )  ,  which  for  the  18  shot  shown  on  the  figure 

would  place  the  mean  point  of  impact  somewhere  near  the  point  P.  Of  course 
the  larger  we  can  make  n,  the  more  accurately  is  the  position  of  the  mean  point  of 
impact  determined.  Then,  the  origin  being  shifted  to  P,  we  get  new  values  of  the 
coordinates  z  and  y,  from  which  we  know  the  deviations  of  each  shot  from  the  mean 
point  of  impact,  both  horizontal  and  vertical.  The  same  process  is  resorted  to  in  the 
horizontal  plane,  with  coordinates  x  and  z  to  determine  the  position  of  the  mean 


Y 


e 


/a  t     Z' 


'y  .  e 


/-f- 


/3 


/z 


Figure  37. 


point  of  impact  in  that  plane,  and  tlience  the  deviations  of  the  several  shots  in  that 
plane. 

386.  Having  found  the  position  of  the  mean  point  of  impact  as  described  above, 
and  the  coordinates  of  the  several  points  of  impact  in  relation  to  it,  we  then  get  the 
mean  dispersion  from  mean  point  of  impact  in  the  lateral  and  in  the  vertical  direc- 
tions by  taking  the  arithmetical  mean  (all  signs  positive)  of  all  the  z  coordinates  for 
the  one  and  of  all  the  y  coordinates  for  the  other,  taking  the  mean  point  of  impact  as 
an  origin.  The  mean  dispersion  from  mean  point  of  impact  in  the  horizontal  plane 
is  determined  in  a  similar  way. 
Probability.  387.  The  probability  of  a  future  event  is  the  numerical  measure  of  our  reason- 

able expectation  that  it  will  happen.  Thus,  knowing  no  reason  to  the  contrary,  we 
assign  an  equal  probability  to  the  turning  up  of  each  of  the  six  different  faces  of  a  die 
at  any  throw,  and  we  may  say  that  the  probability  that  an  ace,  for  example,  will  turn 
up  on  any  single  throw  is  measured  by  the  fraction  ^.  This  does  not  mean  that  we 
should  expect  an  ace  to  turn  up  once  and  once  only  in  every  six  times,  but  merely  that 

in  a  great  number  of  throws,  n,  we  may  reasonably  expect  very  nearly  -—  aces  to  be 

thrown,  and  that  the  greater  n  is  the  more  likely  it  is  that  the  result  will  agree  with 
the  expectation. 


ACCUEACY  AND  PEOBABILITY  OF  GUN  FIRE  239 

388.  If  an  event  may  happen  in  a  ways  and  fail  in  6  ways,  each  of  the  a  +  &  ways 
being  equally  likely  to  occur,  the  probability  that  it  will  happen  is  — ~y  and  the  proba- 
bility that  it  will  fail  is r  ,  and  the  sum  of  these  two  fractions,  unity,  represents  the 

certainty  that  the  event  will  either  happen  or  fail.  Thus,  if  the  probability  that  an 
event  will  happen  be  P,  then  the  probability  that  it  will  not  happen  must  be  1  — P. 
For  example,  since  of  the  52  different  cards  which  may  be  drawn  from  a  pack,  13  are 
spades,  the  probability  that  a  single  card  drawn  from  a  pack  will  be  a  spade  is 

13        1  .  13       39        3 

-^-  =  --  ,  while  the  chance  that  it  will  not  be  a  spade  is  1—  -^^  =  -^k  ==  ^  • 
52        4  52       52        4 

389.  If  the  probability  that  one  event  will  happen  be  P,  and  that  another 
independent  event  will  happen  be  Q,  then  the  probability  that  both  events  will  happen 
will  be  the  product  of  P  and  Q.    For  example,  the  probability  that  a  single  card  drawn 

12 

from  a  pack  will  be  a  face  card  (king,  queen  or  knave)  is  vs  >  and  the  chance  that 

13 

it  will  be  a  spade  is  —  ;  therefore  the  chance  that  it  will  be  either  the  king,  queen  or 

,  .        -,      .    12  ^  13        3 

knave  ot  spades  is  — ^  X  --^  =  -z^  . 
'■  52       52        52 

390.  It  will  be  noted  in  the  preceding  discussion  of  the  laws  of  probability  that 
we  have  been  dealing  with  cases  in  which  one  or  more  of  a  fixed  number  of  events  in 
question  must  either  happen  or  fail;  that  is,  with  definite  numbers  of  equally 
probable  events.  When  we  consider  the  deviations  of  projectiles  this  is  no  longer 
the  case,  for  we  are  then  dealing  with  values  which  may  be  anything  whatever 
between  certain  limits.  We  cannot  assign  any  finite  measure  to  the  probability  that 
a  deviation  shall  have  a  definite  value  because  the  number  of  values  that  it  may  have 
is  unlimited.  With  a  single  throw  of  a  die  there  are  just  six  things  that  may  happen 
and  one  of  the  six  must  happen.  With  the  fire  of  a  gun,  however,  within  the  limits 
which  we  are  considering,  there  are  an  infinite  number  of  points  at  which  the  shot 
may  strike,  and  therefore  no  such  definite  fraction  as  ^  can  be  assigned,  as  was  done 
with  the  die.  We  can,  however,  measure  the  probability  that  the  deviation  of  a 
certain  shot  will  lie  within  certain  limits,  or  that  it  will  be  greater  or  less  than  an 
assigned  quantity.  Suppose,  for  example,  that  a  very  large  number,  n,  of  shots  have 
been  fired,  and  that,  their  lateral  deviations  having  been  measured,  it  is  found  that  m 
of  these  deviations  are  between  2  feet  and  3  feet,  either  plus  or  minus;  then  we 
could  say  that,  in  any  future  trial  under  similar  circumstances,  the  probability  that 

a  single  shot  will  have  a  lateral  deviation  between  2  and  3  feet  is  —  .    Or  if,  of  the  n 

impacts,  q  were  less  than  4  feet  to  one  side  or  the  other  of  the  mean  point  of  impact, 
we  could  say  that  the  probability  that  the  lateral  deviation  of  any  single  shot  would  be 

less  than  4  feet  is  ^  .    The  actual  case  given  in  the  following  paragraph  will  serve  as 

an  illustration,  although  n  is  not  really  as  large  as  it  should  be. 

391.  On  December  17,  1880,  at  Krupp's  proving  ground,  at  Meppen,  50  shots    observed 
were  fired  from  a  12-centimeter  gun  at  5°  elevation,  giving  a  mean  range  of  2894.3 
meters.    The  points  of  fall  were  marked  on  the  ground  and  the  position  of  the  mean 
point  of  impact  was  determined  (note  that  this  is  in  the  horizontal  plane).    Measur- 


240 


EXTERIOR  BALLISTICS 


ing  the  lateral  deviations  from  this  mean  point  of  impact,  the  following  results  were 

obtained: 

_.     .  r Number  of  shot ^ 

•^*™'^^  To  the  right  To  the  left 

Between  0  and  1  meter 1-i  13 

Between  1  and  2  meters 8  8 

Between  2  and  3  meters 2  5 


24 


26 


The  mean  lateral  deviation  was  found  to  be  1.07  meter.  Taking  horizontal  and 
vertical  axes  through  the  mean  point  of  impact  (assuming  that  the  lateral  deviations 
are  the  same  in  the  horizontal  and  in  the  vertical  plane,  which  is  very  nearly  the 
case),  laying  off  equal  spaces  to  left  and  to  right  of  the  origin,  each  representing  one 
meter,  and  constructing  on  each  space  a  rectangle  whose  height  represents,  on  any 

r 


A] 


A. 


b:/ 


/ 


/ 


1^^- 


-\^' 


C'       B'      A'        O        A        B 


Figure  38. 


convenient  scale,  the  number  of  shots  whose  lateral  deviations  were  within  the  limits 

corresponding  to  the  space,  we  obtain  Figure  38. 

392.  It  will  be  seen  that  the  distribution  of  the  deviations  is  fairly  symmetrical 

to  the  axis  of  Y ,  there  being  26  to  the  left  and  24  to  the  right;  also  that  the  maximum 

does  not  exceed  three  times  the  mean  deviation ;  also  that  the  area  of  each  rectangle 

divided  by  the  whole  area  of  the  figure  is  the  measure  of  the  probability  (as  defined) 

that  any  single  deviation  will  fall  within  the  limits  represented  by  its  base.    Thus, 

14 
the  area  OAA^  —  \^,  divided  by  the  total  area,  50,  is  the  probability,  -^  ?  that  any 

single  deviation  will  lie  between  0  and  +1  meter,  the  area  OAA-^A^' A' =  27,  divided 

27 
by  the  total  area  is  the  probability,  -^ ,  that  any  single  deviation  will  lie  between  + 1 

50 
meter  and  —1  meter;  and  the  total  area  divided  by  itself,  -^  =  1  =  certainty,  is  the 

probability  that  no  deviation  will  exceed  3  meters. 


ACCUEACY  AND  PEOBABILITY  OF  GUN  FIEE 


241 


393.  Now  if  the  number  of  shot  be  increased,  while  the  width  of  the  horizontal 
spaces  be  diminished  in  the  same  proportion,  the  area  of  each  rectangle  divided  by 
the  whole  area  of  the  figure  will  continue  to  measure,  with  increasing  accuracy,  the 
probability  that  any  one  deviation  will  fall  within  the  limits  represented  by  its  base. 
At  the  limit,  when  the  number  of  shot  is  infinite  and  the  width  of  the  horizontal 
spaces  has  been  reduced  to  the  infinitesimal  dz,  the  height,  y,  of  each  rectangle  will 
still  be  finite ;  the  upper  contour  of  the  figure  will  become  a  curve  approximately  like 
that  shown  in  the  figure ;  the  area  of  each  rectangle  now  becomes  the  elementary  area 


Mathematical 
theory. 


ydz  and  the  whole  area  under  the  curve  now  becomes 


ydz,  and  the  quotient  of 


the  first  by  the  second  will  still  measure  the  probability  that  any  one  shot  will  fall 
between  the  limits  represented  by  the  base  of  the  rectangle,  that  is,  between  z  and 

z  +  dz.    The  area  between  any  two  ordinates  of  the  curve,  that  is,       ydz,  divided  by 

the  whole  area,  will  still  measure  the  probability  that  any  one  deviation  will  lie 
between  a  and  h. 

394.  The  curve  just  described  is  the  probability  curve  for  the  lateral  deviations 
of  the  projectiles  from  the  particular  gun  considered,  under  the  given  conditions, 
and  while  tlie  probability  curve  for  the  vertical  deviations  for  the  same  case,  or  for 
either  lateral  or  vertical  deviations  or  for  deviations  in  range  in  the  case  of  other  guns 
or  other  conditions  would  difi^er  from  the  particular  curve  shown  in  Figure  38,  they 
would  all  present  the  following  general  features : 

1.  Since  plus  and  minus  deviations  are  equally  likely  to  occur,  the  curve  must 
be  symmetrical  to  the  right  and  to  the  left  of  the  origin,  which  is  the  mean  point  of 
impact. 

2.  Since  the  deviations  are  made  up  of  elemental  deviations  which,  as  they  may 
have  either  direction,  tend  to  cancel  one  another,  small  deviations  are  more  frequent 
than  large  ones,  so  the  maximum  ordinate  occurs  at  the  origin. 

3.  Since  large  deviations  can  only  result  when  most  of  the  elemental  deviations 
have  the  same  directions  and  their  greatest  magnitudes,  such  large  deviations  must  be 
rare,  and  deviations  beyond  a  certain  limit  do  not  occur  at  all.  Therefore  the  curve 
must  rapidly  approach  the  horizontal  axis,  both  to  the  right  and  to  the  left  of  the 
origin,  so  that  the  ordinate,  which  can  never  be  negative,  practically  vanishes  at  a 
certain  distance  from  the  origin. 

395.  If  y  =  F(z)  be  the  equation  to  the  probability  curve,  the  general  features 
stated  in  the  preceding  paragraph  require  that: 

1.  F(z)  shall  be  an  even  function;  that  is,  a  function  of  z'. 

2.  F{0)  shall  be  its  maximum  value. 

3.  It  shall  be  a  decreasing  function  of  z-  and  shall  practically  vanish  when  z  is 
large.  Since  it  is  impracticable  to  so  select  the  function  F  that  F(z)  shall  be  con- 
stantly equal  to  zero  when  z  exceeds  a  certain  limit,  this  last  condition  requires  that  the 
curve  shall  have  the  axis  of  Z  for  an  asymptote;  in  other  words,  we  must  have 
F{±cc)=0. 

396.  The  foregoing  characteristics  being  thus  established,  and  taking  as  a  basis   Probability 

curve* 

the  axiom  that  the  arithmetical  mean  of  the  observed  values  (made  under  similar 
circumstances  and  with  equal  care)  of  any  quantity  is  its  most  probable  value,  the 
theory  of  accidental  errors  deduces  as  the  equation  to  the  probability  curve 

1      -A 

Try 

in  which  y  is  the  mean  error,  or  in  our  case  the  mean  deviation  from  mean  point  of 


Conditions 
existing. 


y= 


(222) 


16 


243 


EXTERIOR  BALLISTICS 


impact,  7r  =  3.1416,  and  e  =  2.7183,  and  the  factor  —  has  been  introduced  to  make  the 
whole  area  under  the  curve  equal  to  unity  '' 

■-f-oo     ___z£_ 

e     Try-  dz  =  Try 


thus  obviating  the  necessity  for  dividing  the  partial  area  by  the  whole  area  whenever 
a  probability  is  to  be  computed. 

397.  Figure  39  represents  the  probability  curve  for  the  Krupp  12-centimeter 
siege  gun,  taking  its  mean  error  to  be  1.07  meter,  as  given  by  the  50  shots  previously 
described.    There  is  also  shown  in  dotted  lines,  for  comparison,  the  probability  curve 


^' 


Figure  39. 


for  a  gun  whose  mean  error  is  three-quarters  that  of  the  12-centimeter  gun.  In  both 
cases  the  ordinates  are  exaggerated  ten  times  as  compared  with  the  abscissa?. 

398.  The  maximum  ordinate  being  the  value  of  y  when  2  =  0,  is  therefore  in- 
versely proportional  to  the  mean  deviation,  that  is,  y         =  —  ;  the  probability 

that  any  one  deviation  will  be  less  than  OB  =  OA  is  the  numerical  value  of  the  area 
AA'CB'B  in  the  one  case,  and  of  the  area  AA'C'B'B  in  the  other;  the  probability  that 
any  one  deviation  will  exceed  OB  =  OA  is  the  area  under  that  part  of  the  curve  which 
is  to  the  left  of  AA'  and  to  the  right  of  BB' ;  the  whole  area  under  the  curve  has  the 
numerical  value  of  unity.  It  will  be  seen  how  very  small  is  the  probability  that  any 
deviation  will  exceed  three  times  the  mean  deviation. 


ACCURACY  AND  PROBABILITY  OF  GUX  FIRE 


243 


399.  The  probability,  P,  that  the  deviation  of  any  single  shot  will  be  numerically 
less  than  a  given  quantity,  a,  being  measured  by  the  area  between  the  ordinates  of  the 
probability  curve  at  2=  dza^  and  that  curve  being  symmetrical  to  the  axis  of  Y,  we  have 


Try  Jo 


dz 


(223) 


400.  In  order  to  avoid  repeated  integrations,  the  following  table  gives  the  value 
of  P,  calculated  from  the  above  equation,  but  arranged  for  convenient  use  with  the 

ratio  —  as  an  argument.     Knowing  the  mean  deviation  of  a  gun,  y,  to  find  the 

probability  of  a  shot  striking  within  a  given  distance  of  the  mean  point  of  impact,  it  is 

only  necessary  to  take  from  the  table  the  value  of  P  which  corresponds  to  — .    It  is  to 

be  noted  that  if  a  and  y  relate  to  lateral  errors  on  a  vertical  screen,  we  get,  by  the  use 
of  this  table,  the  probability  that  any  one  shot  will  strike  between  the  two  vertical 
lines  on  the  screen  distant  a  to  the  right  and  left,  respectively,  of  the  mean  point  of 
impact  on  the  vertical  screen ;  that  if  a  and  y  relate  to  vertical  errors  on  a  vertical 
screen,  we  get  the  probability  that  any  one  shot  will  strike  between  two  horizontal 
lines  distant  a  above  or  below  the  mean  point  of  impact,  respectively ;  if  a  and  y  relate 
to  the  point  of  impact  in  the  horizontal  plane  and  to  lateral  deflections,  we  get  the 
probability  that  any  single  shot  will  fall  between  two  lines  drawn  on  the  surface  of 
the  water  parallel  to  the  horizontal  trace  of  the  vertical  plane  of  the  mean  trajectory 
and  distant  a  to  the  right  and  left,  respectively,  from  the  mean  point  of  impact ;  and 
if  a  and  y  relate  to  the  point  of  impact  in  the  horizontal  plane  and  to  deviations  in 
range,  we  get  the  probability  that  any  single  shot  will  fall  between  two  lines  drawn 
on  the  surface  of  the  water  perpendicular  to  the  horizontal  trace  of  the  vertical  plane 
of  the  mean  trajectory  and  a  short  of  or  beyond  the  mean  point  of  impact.  Each  one 
of  these  four  cases  is  of  use  under  proper  conditions. 

PROBABILITY  OF  A  DEVIATION  LESS  THAN  a  IN  TERMS  OF 
THE  RATIO  —  . 


a 

a 

a 

a 

P. 

P. 

P. 

P. 

7 

7 

7 

7 

O.I 

.004  " 

1.1 

.620 

2.1 

.906 

3.1 

.987 

0.2 

.127 

1.2 

.662 

2.2 

.921 

3.2 

.990 

0.3 

.189 

1.3 

.700 

2.3 

.934 

3.3 

.992 

0.4 

.2.50 

1.4 

.735 

2.4 

.945 

3.4 

.994 

0.5 

.310 

1.5 

.768 

2.5 

.954 

3.5 

.995 

0.6 

.368 

1.5 

.798 

2.6 

.962 

3.6 

.996 

0.7 

.424 

1.7 

.825 

2.7 

.969 

3./ 

.997 

0.8 

.477 

1.8 

.849 

2.8 

.974 

3.8 

.998 

0.9   , 

.527 

1.9 

.870 

2.9 

.979 

3.9 

.998 

I.O 

.575 

2.0 

.889 

3.0 

.983 

4.0 

.999 

401.  x\s  an  illustration  of  the  use  of  the  above  table,  we  will  find  the  probability 
of  a  deviation  not  exceeding  1  meter  and  2  meters  in  the  case  of  a  gun  whose  mean 
lateral  deviation  is  1.07  meter,  and  will  compare  our  results  with  those  given  by  the 
actual  firing  of  50  shots  from  the  Krupp  12-centimeter  gun.    Taking  a  =  l  meter,  we 

have  —  =  — -— ;  =,935,  and  from  the  table  P=.544.    The  probability  that  the  lateral 

deviation  will  not  exceed  1  meter  is  therefore  .544;  therefore  of  50  shots  27  should 
fall  within  1  meter  on  either  side  of  the  mean  point  of  impact,  and  actually  27  did  so 


fall.    Taking  a  =  2  meters,  we  have 


-— —  =1.87,  whence  P  =  .864,  which  is  the 


244  EXTERIOR  BALLISTICS 

probability  that  the  lateral  deviation  of  any  one  shot  will  not  exceed  2  meters. 
Therefore  of  50  shots  43  should  fall  within  2  meters  on  either  side  of  the  mean  point 
of  impact,  and  actually  43  did  so  fall. 

402.  If  P  be  the  probability  that  the  deviation  of  any  single  shot  will  not  be 
greater  than  a,  then  evidently  lOOP  will  be  the  probable  number  of  shots  out  of  100 
which  will  fall  within  the  limits  ±a;  in  other  words,  lOOP  is  the  percentage  of  hits 
to  be  expected  upon  a  band  2a  wide  with  its  center  at  the  mean  point  of  impact. 
Thus  we  see  from  the  table  that  the  half  width  of  the  band  which  will  probably 
receive  25  per  cent  of  the  shot  is  0.4y,  while  the  half  width  of  the  band  that  will  prob- 
ably receive  50  per  cent  of  the  shot  is  0.846y.  These  facts  are  usually  expressed  by 
saying  that  the  width  of  the  25  per  cent  rectangle  is  0.80  and  of  the  50  per  cent 
rectangle  is  1.69  times  the  mean  error. 

403.  The  half  width  of  the  50  per  cent  rectangle  is  known  as  the  "  probable 
error,"  or  in  our  case  the  "  probable  deviation,"  since  it  is  the  error  or  deviation  which 
is  just  as  liable  to  be  exceeded  as  it  is  not  to  be  exceeded. 

404.  If  we  wish  to  find  the  probability  of  hitting  an  area  whose  width  is  2&  and 
whose  height  is  2h,  since  the  lateral  and  vertical  deviations  are  independent  of  each 
other,  the  probability  is  the  product  of  the  two  values  of  P  taken  from  the  table  with 

the  arguments  —  and  — -  ,  where  y«  and  vy  are  the  mean  lateral  and  mean  vertical 

deviations,  respectively.  Thus,  supposing  jz  to  be  4  feet  and  jy  to  be  5  feet,  the 
probability  of  hitting  with  a  single  shot  a  20-foot  square  with  its  center  at  the  mean 
point  of  impact  is  PiP2  =  .954x  .889  =  .848,  Pi  =  .954  being  the   value  of  P  for 

-^  =  ^  =2.5  and  P.  =  -889  being  the  value  of  P  for  A  =  ^  =2. 


EXAMPLES. 

1.  The  coordinates  {z,  y)  of  10  hits  made  by  a  6-pounder  gun  on  a  vertical 
target  at  2000  yards  range,  axes  at  center  of  target,  were  as  follows,  in  feet : 

(-10,  +13)  (  +  11,  +9)  (+4,  -2)  (-1,  +1) 
(-  4,  +  2)  (+  2,  +1)  (-1,  -2)  (  0,  -4) 
(-    1,   -    3)      (-   4,   -4) 

Find  the  mean  point  of  impact  and  the  mean  vertical  and  lateral  deviations. 

Answers.     Zq=—QA;  ^^=+1.1;  yj/  =  4.14;  y~  =  3.72. 

2.  The  coordinates  of  8  hits  made  by  a  28-centimeter  gun  on  a  vertical  target  at 
4019  meters  range,  axes  at  center  of  target,  were  as  follows,  in  centimeters : 

(-80,   -90)      (-    10,   +210)      (  +  30,   -70)      (-70,   +355) 
(  +  30,   +40)      (-220,   -150)      (-40,   +40)      (-65,   +    90) 

Find  the  mean  j^oint  of  impact  and  the  mean  vertical  and  lateral  deviations. 

Answers.     z^i=—bd;  ^^=+53;  y^  =  123.9;  yj  =  55.7 


ACCURACY  AND  PEOBABILITY  OF  GUN  FIRE  245 

3.  The  following  ranges  and  lateral  deflections  from  the  plane  of  fire,  in  meters, 
were  given  by  10  shots  from  a  28-centinieter  gun  at  8°  30'  elevation: 

Range.  Deflection  left.  Range.  Deflection  left. 

6285                          18  6204                         16 

6228                         21  6141                         17 

6187                        15  6200                        19 

6187                         13  6256                         15 

6192                         17  6205                         18 

Find  the  mean  point  of  impact,  the  mean  lateral  deviation  and  the  mean  deviation 
in  range. 

Ansivers.     Mean  range  6208.5  meters  ya;  =  28.7  meters. 

Mean  lateral  deflection  16.8  meters     y-=    1.8  meters. 

4.  A  and  B  shoot  alternately  at  a  mark.  If  A  can  hit  once  in  n  trials  and  B 
once  in  n  —  1  trials,  show  that  their  chances  are  equal  for  making  the  first  hit.  What 
are  the  odds  in  favor  of  B  after  A  has  missed  the  first  shot? 

Answer,     n  to  n  —  2. 

5.  What  is  the  probability  of  throwing  an  ace  with  a  single  die  in  two  trials? 

Answer.      777^ . 
3o 

6.  Taking  the  mean  vertical  error  given  from  Example  1,  and  supposing  the 
mean  point  of  impact  to  be  at  the  center  of  a  vertical  target,  what  would  be  the  per- 
centage of  hits  on  targets  of  unlimited  width  and  of  heights,  respectively,  of  8  feet, 
12  feet,  16  feet,  20  feet  and  24  feet? 

Anstvers.     55.9;i;;  75.1^;  87.6^;  94.6^;  97.9^. 

7.  Taking  the  mean  errors  given  from  Example  1,  what  percentage  of  shot  would 
enter  a  gun  port  4  feet  square,  supposing  the  mean  point  of  impact  to  be  at  the  center 
of  the  port?  What  would  be  the  percentage  if  the  port  were  3  feet  high  by  5  feet 
wide?  Answers.     9.9^;  9.2^. 

8.  What  would  be  the  probability  of  a  single  shot  from  the  28-centimeter  gun  of 
Example  2  hitting  a  turret  2  meters  high  and  8  meters  in  diameter  at  the  range  for 
which  the  mean  errors  are  given,  supposing  the  fire  to  be  accurately  regulated  ? 

Answer.     0.48. 

9.  If  a  zone  of  a  certain  width  receives  20^  of  hits,  how  many  times  as  wide  is 
the  zone  which  receives  80;^  of  hits?  Answer.     5.05  times. 

10.  At  Bucharest,  in  1886,  94  shots  were  fired  from  a  Krupp  21-centimeter  rifled 
mortar  at  a  Gruson  turret,  distant  2510  meters,  without  hitting  it.  The  mean  devia- 
tions were  33.27  meters  in  range  and  9.90  meters  laterally,  and  the  mean  point  of 
impact  practically  coincided  with  the  center  of  the  turret.  What  was  the  probability 
of  hitting,  supposing  the  target  to  have  been  a  6-meter  square  (it  was  really  a  circle 
of  6  meters  diameter)  ?  Answer.     0.011. 

11.  How  many  of  the  94  shots  of  Example  10  would  probably  have  struck  a 
rectangle  80  meters  by  16  meters,  with  the  longer  axis  in  the  plane  of  fire? 

Answer.     '31.8i. 


CHAPTER  21. 

THE  PROBABILITY  OF  HITTING  WHEN  THE  MEAN  POINT  OF  IMPACT  IS 
NOT  AT  THE  CENTER  OF  THE  TARGET.  THE  MEAN  ERRORS  OF  GUNS. 
THE  EFFECT  UPON  THE  TOTAL  AMMUNITION  SUPPLY  OF  EFFORTS  TO 
SECURE  A  GIVEN  NUMBER  OF  HITS  UPON  A  GIVEN  TARGET  UNDER 
GIVEN  CONDITIONS.  SPOTTING  SALVOS  BY  KEEPING  A  CERTAIN  PRO- 
PORTIONATE NUMBER  OF  SHOTS  AS  "  SHORTS." 

Mean  point  405.  In  the  preceding  chapter  we  considered  only  the  chance  of  hitting  when 

not°It'?^nter  the  mean  point  of  impact  is  at  the  center  of  the  target,  but  this  is  far  from  being  an 
of  target.  ^|.^^-j^g^|^jg  Condition  in  the  service  use  of  guns,  especially  of  naval  guns.  In  fact  to 
bring  the  mean  point  of  impact  upon  the  target  is  the  main  object  to  be  attained  in 
gunnery,  for,  from  what  has  already  been  said,  if  the  mean  point  of  impact  be 
brought  into  coincidence  with  the  center  of  the  target  and  kept  there,  we  will  get  the 
maximum  number  of  hits  possible,  and  it  is  to  the  accomplishment  of  this  that  the 
spotter  gives  his  efforts.  Even  with  a  stationary  target,  at  a  known  range,  however, 
it  is  difficult  to  so  regulate  the  fire  as  to  bring  about  and  maintain  this  coincidence 
of  center  of  target  and  mean  point  of  impact;  and  when  the  target  is  moving  with  a 
speed  and  in  a  direction  that  are  only  approximately  known ;  when  the  range  is  not 
accurately  known;  when  there  is  a  wind  blowing  which  may  vary  in  force  and  direc- 
tion at  different  points  between  the  gun  and  the  target;  when  the  density  of  the  air 
may  vary  at  different  points  between  the  gun  and  the  target;  and  when  the  firing 
ship  is  also  in  motion,  etc. ;  even  the  most  expert  regulation  of  the  fire  by  the  observa- 
tion of  successive  points  of  fall  can  do  no  more  than  keep  the  mean  point  of  impact 
in  the  neighborhood  of  the  object  attacked.  All  this  applies  to  a  single  gun,  and  in 
salvo  firing  we  have  the  additional  trouble  that  the  mean  points  of  impact  of  the 
several  guns  cannot  be  brought  into  coincidence.  This  makes  it  necessary  for  the 
spotter  to  estimate  the  position  of  the  mean  point  of  impact  of  the  whole  salvo,  that  is, 
the  mean  position  of  the  mean  points  of  impact  of  all  the  guns,  and  it  is  this  com- 
bined mean  point  of  impact  of  all  the  guns  that  the  spotter  must  determine  in  his 
own  mind  and  endeavor  to  bring  upon  the  target  and  keep  there.  The  difficulties 
attending  this  process  are  manifest. 

406.  In  Figure  40,  let  0  be  the  mean  point  of  impact  of  a  single  gun,  and  let 
ABCD  be  the  target  at  any  moment,  and  let  the  coordinates  of  the  center  of  ABCD, 
with  reference  to  the  horizontal  and  vertical  axes  through  0  be  z^  and  ?/o ;  also  let  the 
mean  lateral  and  vertical  deviations  of  the  gun  be  y^  and  jy,  respectively,  and  let  the 
dimensions  of  the  target  be  2&  and  21i.  Then  the  probability  that  a  shot  will  fall 
between  the  vertical  lines  Cc  and  C'c'  is  the  tabular  value  of  P  for  the  argument 

?^i^  ,  which  we  will  call  Pz{z^  +  h)  ;  and  the  probability  that  a  shot  will  fall  between 

^^'                                                                                              z  -b 
Dd  and  D'd'  is  the  tabular  value  of  P  for  the  argument  -^ ,  which  we  will  call 

7z 

P^(zo-h).     Therefore  the  probability  that  a  shot  will  fall  between  Cc- and  Dd  is 
one-half  the  difference  of  the  two  preceding  probabilities,  or 

^[P,(z,  +  h)-P.iZo-h)]  (224) 

Similarly,  the  probability  that  a  shot  will  fall  between  the  horizontal  lines  C'C  and 
^'i^is 

i[Pyiyo+h)-PAyo-h)]  (225) 


ACCUEACY  AND  PEOBABILITY  OF  GUN  FIEE 


247 


Hence  the  probability  of  hitting  ABCD  is  the  product  of  the  two  expressions  given 
in  (224)  and  (225),  or 

i[P^{^^o  +  ^)-PAz,-h)-\x[PAu,  +  h)-Py{ij,-h)]  (226) 

T 


c' 


7>' 


3' 


^' 


I- 


^. 


A' 


O 


\a.' 


-f 


d' 


Figure  40. 


1 
I 


^ 


7c 
I 

jr. 


Zh. 

B 


407.  To  illustrate,  suppose  we  wish  to  find  the  probable  percentage  of  hits  on  a 
gun  port  4  feet  square,  if  the  mean  point  of  impact  be  3  feet  to  one  side  of  and 
4  feet  below  the  center  of  the  port,  the  value  of  yz  being  3.72  feet  and  of  yj/  being 
4.14  feet.    Here  we  have : 

P,(2,+  &)=F,(5)=.717  P,(;,/„  +  /0=P,(G)r:=.751 

F,{z,-l)  =P,il)  =.Vm  Pyiyo-h)  =P,j{2)  =:.298 

Pz{o)  -P.(l)  =;547  P,(0)  -P,(2)  =.453 

From  which  we  have  P  =  ^x  .547  x  .453  =  .062 

Therefore  the  percentage  of  hits  under  the  given  conditions  would  be  6.2  per  cent. 
Under  the  same  conditions,  but  with  the  mean  point  of  impact  at  the  center  of  the 
port,  the  percentage  of  hits  would  be  9.9  per  cent. 

408.  From  what  has  been  said  it  is  evident  that  the  less  the  mean  errors  of  the 
gun,  the  more  important  it  becomes  to  accurately  regulate  the  fire;  for  if  the  dis- 
tance of  the  mean  point  of  impact  from  the  target  be  more  than  three  times  the 
mean  error  of  the  gun  we  would  get  practically  no  hits  at  all.  Therefore  a  reduction 
in  the  mean  error  of  the  gun  renders  imperative  a  corresponding  reduction  in  the 
distance  within  which  the  spotter  must  keep  the  mean  point  of  impact  from  the 
target  if  hits  are  to  be  made.  Therefore,  unless  good  control  of  the  fire  be  secured,- 
a  gun  with  a  small  mean  error  will  make  fewer  hits  than  one  with  a  larger  mean 
error,  and  this  has  sometimes  been  used  as  an  argument  in  favor  of  guns  that  do  not 


Bearing:  of 
mean  errors 
upon  fixe 
control. 


248 


EXTERIOR  BALLISTICS 


shoot  too  closely.  Conversely,  however,  if  good  control  be  secured — that  is,  if  the 
spotter  be  competent  and  careful — the  close-shooting  gun  will  secure  more  hits  than 
the  other.  Therefore  the  scientific  method  of  securing  hits  is  to  have  a  competent 
spotter  and  a  close-shooting  gun;  the  other  process  is  a  discarding  of  science  and 
knowledge  and  a  falling  back  upon  luck,  which  cannot  but  meet  with  disaster  in  the 
face  of  an  enemy  using  proper  and  scientific  methods. 

409.  To  illustrate  the  statements  contained  in  the  preceding  paragraph,  we  will 
take  the  case  of  a  6"  gun  firing  at  a  turret  25  feet  high  by  32  feet  in  diameter,  and 
3000  yards  distant.  Suppose  the»mean  vertical  and  lateral  deviations  each  to  be  10 
feet;  if  the  mean  point  of  impact  coincides  with  the  center  of  the  target,  the  probable 
percentage  of  hits  will  be  54.3  per  cent;  but  if  the  sights  be  set  for  a  range  of  10  per 
cent  more  or  less  than  the  true  distance  the  mean  point  of  impact  will  be  raised  or 
lowered  about  43  feet  (this  is  one  of  the  older  6"  guns;  not  the  one  given  in  the 
accompanying  range  tables),  and  the  percentage  of  hits  will  be  reduced  to  0.7  per 
cent.  If,  on  the  other  hand,  the  mean  errors  of  the  gun  were  each  20  feet,  or  double 
the  first  assumption,  while  the  percentage  of  hits  with  perfect  regulation  of  fire 
(that  is,  with  the  mean  point  of  impact  at  the  center  of  the  target)  would  be  reduced 
to  18.2  per  cent,  that  with  sight  setting  for  a  range  10  per  cent  in  error  would  be 
4.7  per  cent.  Thus  we  see  that  if  the  fire  be  not  accurately  regulated  a  gun  will  be 
severely  handicapped  by  its  own  accuracy  if  the  range  be  not  known  within  10 
per  cent. 

410.  The  work  for  the  problem  in  the  preceding  paragraph  is  as  follows : 


Figure  41. 


Case  1.    Mean  deviation  10  feet.    Mean  point  of  impact  at  0  (Figure  41), 

a^         16 


10 
L2J 
10 


=  1.6 


:.798 


2  =1^=1.25         P,,  =  .681 


P,xPy  =  . 543438 


Therefore  the  percentage  of  hits  is  54.3  per  cent. 


ACCUEACY  AND  PEOBABILITY  OF  GUN  FIEE 


249 


Case  2.    Mean  deviation  10  feet.    Mean  point  of  impact  at  0  (Figure  42), 
Chances  of  hitting  between  A'D  and  B'C. 

P,(2o  +  16)-P.(.-o-16)=F^(lG)-P.(-16)  =1.596 

P,(16)=.T98 


.0  =  0         ^-  —  =1.6 


Chances  of  hitting  between  .45  and  CD. 

Py{y,  +  12.h)  -P,(,y,-12.5)  =P,(55.5)  -P,(30.5) 

^-43-^  =-^=5.55  P,(55.5)  =1.0000 
-^^  y^      10 


=  .0168 
i  I  .026208 
.0065 


■^3  _ 


30.5 
10 


=  3.05  P2,(30.5)=  .9832 
.0168 


Percentage  of  hits  is  0.7  of  1  per  cent. 

Case  3.    Mean  deviation  20  feet.    Mean  point  of  impact  at  0  (Figure  41). 

P,  =  .477 

Pj,  =  .382 


«1 

= 

16 
20 

=  0.8 

'yy 

= 

12.5 

20 

=    .62 

5 

P«xPj,=  . 182214 
Percentage  of  hits  is  18.2  per  cent. 

Case  4.    Mean  deviation  20  feet.    Mean  point  of  impact  at  0  (Figure  42). 

Y 


<-r-3Z 

1 

1 
1 

> 

1                           /N 
ZS'                      ' 

i       i 

< /6' > 

3' 


f3' 


A 


O 


A' 


Figure  42. 

Chances  of  hitting  between  A'D  and  B'C. 

P,(2o  +  16)-P.(2o-16)=P^(lC)-P.(-16)  = 

gj   _  1^  —  8 

y.      20    • 


.954 


250  EXTEEIOR  BALLISTICS 

Chances  of  hitting  between  AB  and  CD. 

Py{y,  +  12.5)  -Py(y^-12.5)  =Py(55.D)  -P.,(30.5)  =  .19725 

o^        55.5      ..  .^.      .,..-_.       ._..  4  I  .1881765 


.^Q   -2.775     Pj,(55.5)  =.97275  ^-^ 


:^  =  ^  =  1.525     P^(30.5)  =.77550 


Pj,(55.5)  -Fj,(30.5)  =.19725 
Percentage  of  hits  is  4.7  per  cent. 

411.  If  we  know  the  percentage  of  hits  at  a  given  range  on  a  target  of  given  size, 
we  can  make  a  rough  estimate  of  the  mean  errors  of  the  gun  by  assuming  that  the 
mean  point  of  impact  was  at  the  center  of  the  target,  and  the  greater  the  number  of 
rounds  fired  the  more  nearly  correct  will  this  determination  probably  be.  For 
example,  on  a  certain  occasion,  the  eighty  6"  gun  of  certain  British  ships,  firing 
separately,  made  295  hits  out  of  650  rounds  fired;  that  is,  45.4  per  cent  of  hits ;  on 
a  target  15  feet  high  by  20  feet  wide,  at  a  mean  range  of  1500  yards.  Here  we  have 
given  that  the  product 


-^(f)x-»a=-- 


Assuming  that  yz  =  yy,  we  may  solve  the  above  by  a  process  of  trial  and  error,  that  is, 
by  assuming  successive  integral  values  of  y,  and  by  this  process  we  see  that  when 
y  =  7  we  have 

^)  X  Py  (Jy)  =  -^45  X  .581  =  .432 

and  as  .432  is  very  nearly  .454,  we  can  say  that  the  mean  deviations  are  slightly  less 
than  7  feet;  and  we  could  go  on  and  determine  the  solution  of  the  equation  more 
accurately  by  trying  6.9  feet  instead  of  7  feet  as  the  value  of  y.  This  would  probably 
not  make  the  result  any  nearer  the  truth,  however,  as  any  correction  resulting  there- 
from would  probably  be  less  than  the  error  caused  by  the  assumption  that  the  mean 
point  of  impact  was  at  the  center  of  the  target. 

412.  The  number  of  roimds  necessary  to  make  at  least  one  hit  may  be  determined 
by  the  following  method:  Let  p  be  the  probability  of  hitting  with  a  single  shot; 
then  1  —  ;^  is  the  probability  that  a  single  shot  will  miss;  and  (1  — ;j)"  is  the  prob- 
ability tliat  all  of  n  shots  will  miss.  Therefore  the  probability  of  hitting  at  least 
once  with  n  shots  is  P=l—  (1  — p)".    Solving  this  equation  for  n,  we  get 

log(l— jP)  =n  log(l  — p) 

log(l-p) 
and  by  giving  P  a  value  near  unity  we  can  find  the  value  of  n  which  will  make  one 
hit  as  nearly  certain  as  we  wish. 

413.  As  an  example,  taking  a  case  in  which  94  shots  were  fired  from  a  mortar  at 
a  turret,  and  in  which  the  calculated  probability  of  a  hit  with  a  single  shot  was  .011, 
let  us  see  how  many  rounds  would  have  to  be  fired  to  make  the  probability  of  at  least 
one  hit  .95.    In  that  case,  /j  =  .011,  and  so  we  have,  from  (227), 

_    log(l-.95)    _   log  .05  _  8^69897_^10  _  -1.3Q1Q3  _^„. 
^      log (1-. Oil)       log  .989       9.99520-10       -0.00480 

Therefore  271  shots  must  be  fired  to  make  the  odds  19  to  1  that  there  will  be  at  least 
one  hit.    The  probability  of  at  least  one  hit  with  the  94  shots  fired  was 

P=l-(l-/;)'*^  =  l-.354  =  .646 


ACCUEACY  AXD  PEOBABILITY  OF  GUN  FIRE  251 

414.  Tlie  deviations  of  the  projectiles  fired  from  a  gun  on  a  steady  platform, 
the  mean  of  which  we  will  call  the  mean  error  of  the  gun,  lateral  or  vertical  as  the 
case  may  be,  are  principally  caused  by : 

1.  Errors  of  the  gun  pointer  in  sighting  the  gun,  but  bear  in  mind  that  this 
does  not  include  "  mistakes,"  which  are  supposed  to  have  been  eliminated,  but  only 
accidental  errors  that  must  necessarily  ensue  even  when  the  pointer  is  working  as 
accurately  as  it  is  humanly  possible  to  do. 

2.  An  initial  angular  deviation  of  the  projectile;  that  is,  when  the  projectile 
does  not  leave  the  muzzle  in  a  path  in  line  with  the  axis  of  the  gun. 

3.  Variations  in  initial  velocity  between  successive  rounds. 

4.  Differences  between  projectiles  existing  even  after  all  possible  differences 
have  been  eliminated. 

415.  With  open  sights  the  most  expert  gun  pointers  make  a  considerable  angular 
error  (this  is  an  angular  error  in  sighting,  not  to  be  confused  with  the  angular 
deviation  described  in  2  of  the  preceding  paragraph),  which  varies  from  round  to 
round,  when  the  gun  is  pointed  by  directing  the  line  of  sight  at  a  target.  With 
telescopic  sights  this  error  is  greatly  reduced  but  still  exists.  There  is  also  always 
an  error  in  setting  the  sights  (we  suppose  the  range  to  be  unchanged  from  shot  to 
shot,  but  that  the  sights  are  reset  for  each  round).  The  mean  angular  error  of 
sighting  can  only  be  estimated. 

416.  The  initial  angular  deviation  results  from  the  projectile  not  leaving  the  gun 
in  the  exact  line  of  the  axis  of  the  latter.  This  deviation,  which  occurs  indifferently 
in  all  directions,  was  quite  large  with  smooth-bore  guns  and  with  some  of  the  earlier 
rifles,  but  with  modern  guns,  using  projectiles  rotated  by  forced  bands,  it  is  un- 
doubtedly much  less. 

417.  With  all  powders  the  muzzle  velocity  varies  somewhat  from  round  to  round, 
no  matter  what  care  be  taken  to  insure  uniformity  in  the  charges.  With  the  nitro- 
cellulose powder  now  used  in  our  navy,  if  the  charges  have  been  made  up  with 
proper  care,  and  if  the  projectiles  are  all  of  the  same  weight,  the  average  difference 
between  the  velocities  given  by  successive  rounds  and  the  mean  velocity  of  all  the 
rounds  fired  on  any  one  occasion  will  prol)ably  not  be  great.  If  a  large  number  of 
rounds  be  fired,  and  the  velocity  for  no  one  round  differs  a  f .  s.  from  the  average,  then 

somewhat  more  than  half  the  velocities  will  be  within  -;-  f .  s.  of  the  average,  a  not 

o 

being  large. 

418.  The  projectiles  of  any  gun  differ  among  themselves,  but  when  they  all  have 
the  same  form  of  head  and  are  not  of  greatly  different  lengths,  the  resulting  devia- 
tions are  not  so  important  as  those  caused  by  variations  in  the  weight. 


252 


EXTERIOE  BALLISTICS 


419.  Of  course  the  only  correct  way  of  determining  the  mean  errors  of  a  given 
gun  is  by  actually  firing  a  large  number  of  rounds  at  a  target  and  measuring  the 
deviations.  That  the  errors  are  very  small  under  favorable  circumstances  is  illus- 
trated in  Figure  43,  vi^hich  represents  a  target  made  at  Meppen  on  June  1,  1882,  with 
a  28-centimeter  gun,  the  distance  of  the  target  from  the  gun  being  2026  meters  (2215 
yards).  The  dotted  cross  is  the  mean  point  of  impact,  whose  coordinates  referred  to 
the  horizontal  and  vertical  axes  at  the  center  of  the  target  are  2  =  32.4  inches  and 
y=—11.6  inches.  The  mean  lateral  deviation  is  9.5  inches,  and  the  mean  vertical 
deviation  is  11.6  inches.  These  actual  deviations  are  considerably  less  than  those 
encountered  in  service,  which  may  be  plausibly  ascribed  to  the  fact  that  in  proving 


I 
I 
I 
I 
I 
I 
I 
1 
I 


I 
I 
I 
I 
I 
I 


^:s^ >j 


1 

Figure  43. 


ground  firings  greater  care  can  be  taken  in  pointing  than  is  usually  practicable  under 
service  conditions. 
Effect  of  420.  The  three  angular  motions  of  a  ship's  deck,  caused  by  the  rolling,  pitching 

pitching  and   and  yawing,  greatly  increase  the  actual  mean  errors  of  naval  guns  in  service,  but 

y&wing,  ,  o  -^ 

their  effects  depend  so  much  upon  the  skill  of  the  gun  pointer,  as  well  as  upon  the 
state  of  the  sea  and  the  characteristics  of  the  particular  ship  and  gun  mounting,  that 
only  the  roughest  estimates  of  their  values  can  be  made.  Many  naval  guns  are 
mounted^  in  broadside  and  only  train  from  bow  to  quarter,  and  even  those  mounted 
on  the  midships  line  are  likely  to  be  most  used  in  broadside;  thus  the  roll,  which  is 
the  greatest  and  most  rapid  of  a  ship's  motions,  has  its  largest  component  in  the 
plane  of  fire,  and  acts  principally  to  increase  the  vertical  deviations.  The  principal 
effect  of  pitching,  on  the  other  hand,  is  to  increase  the  lateral  deviations  by  causing 
the  plane  of  the  sights  to  be  more  or  less  inclined,  now  to  one  side  and  now  to  the 
other  of  the  plane  of  fire.  Motion  in  azimuth,  yawing,  mostly  due  to  unsteady 
steering,  affects  the  lateral  deviations  only. 

421.  If  the  target  be  in  motion,  the  person  controlling  the  fire  of  the  gun  must 
of  course  estimate  its  speed  and  direction  in  order  to  direct  the  fire  at  the  point 
where  the  target  will  be  when  the  projectile  strikes,  and  his  corrections  must  always 
vary  in  accuracy  from  round  to  round,  thus  increasing  both  the  lateral  and  ihe 
vertical  deviations.  Furthermore,  variations  in  the  accuracy  of  the  estimated  cor- 
rections for  the  speed  of  the  firing  ship  and  for  the  effect  of  the  wind  must  occur 


Uotion  of 
target. 


errors. 


ACCURACY  AND  PROBABILITY  OF  GUN"  FIRE  253 

from  round  to  round,  which  will  also  affect  the  lateral  and  the  vertical  deviations. 
Moreover,  the  changing  direction  of  the  target  will  cause  the  angle  between  the  direc- 
tion of  the  wind  and  the  plane  of  fire  to  vary,  thus  necessitating  a  variable  allowance 
for  wind  effect  and  again  increasing  the  deviations  of  projectiles. 

422.  Taking  everything  into  account,  probably  a  fair  estimate  of  the  mean 
vertical  deviation  of  modern  naval  guns  of  medium  and  large  caliber,  at  2000  yards 
range,  with  skilled  fire-control  personnel  and  gun  pointers,  and  under  average  con- 
ditions, would  be  5  feet.  The  mean  lateral  deviation,  which  for  guns  on  steady 
platforms,  is  from  two-thirds  to  three-fourths  the  mean  vertical  deviation,  may  be 
taken  to  be  the  same  as  the  mean  vertical  deviation  in  the  case  of  naval  guns,  without 
any  great  error.  Both  vertical  and  lateral  deviations  may  be  taken  to  be  proportional 
to  the  range,  at  least  up  to  4000  or  5000  yards  range,  though  the  former  really  in- 
creases somewhat  more  rapidly  than  the  range.  For  ranges  greater  than  those  given, 
the  increase  in  the  deviations  will  be  at  a  greater  rate. 

423.  It  will  be  noticed  that,  in  the  earlier  part  of  the  discussion  of  the  subject,  inherent 
we  referred  to  the  "  accidental  deviations  "  of  a  gun  as  being  due  to  the  "  inherent 
errors  "  of  the  gun,  but  we  have  now  seen  that  there  are  "  accidental  errors  "  that 
are  not  really  inherent  in  the  gun  itself,  although  their  results  are  similar.  The 
angular  deviation  resulting  from  the  fact  that  the  shell  does  not  leave  the  gun  exactly 
in  the  axis  of  the  gun  is  strictly  an  inherent  error  of  the  gun ;  but  the  angular  error 
in  pointing  due  to  the  fact  that  even  the  most  perfectly  trained  and  most  skillful 
pointer  cannot  point  twice  exactly  alike  is  an  error  that  does  not  pertain  to  the  gun 
itself  but  to  its  manipulation.  The  results,  as  stated,  are  similar,  however,  and  may 
therefore  be  considered  together,  as  making  up  the  sum  of  the  accidental  errors  that 
cause  the  deviations.  To  recapitulate,  we  have  first  deviations  due  to  mistakes,  which 
we  eliminate  from  consideration.  Then  M^e  have  deviations  resulting  from  known 
causes,  which  we  also  eliminate  by  the  methods  of  exterior  ballistics.  When  these 
two  sources  of  error  have  been  eliminated,  we  have  remaining  two  sources  of  error, 
those  pertaining  to  the  imperfections  of  the  gun  itself  and  those  pertaining  to  the 
inherent  imperfections  of  even  the  most  perfect  personnel  handling  the  gun.  It  is 
these  last  two  only  that  may  be  considered  under  the  theory  of  probability.  And  bear 
in  mind  the  difference  between  a  mistake  and  an  accidental  error.  A  mistake  is  the 
result  of  bad  judgment,  and  may  cause  a  large  error,  as,  say,  a  mistaken  estimate  of 
two  points  in  the  direction  of  the  wind;  and  an  accidental  error  is  that  small  error 
which  must  necessarily  be  made  even  by  a  thoroughly  trained  judgment.  Accidental 
errors  are  necessarily  small,  and  are  necessarily  as  likely  to  occur  on  one  side  as  on 
the  other. 

424.  Although  the  targets  of  naval  guns  are  generally  vertical,  the  fire  of  such 
guns  must,  as  a  rule,  be  regulated  by  the  observation  of  points  of  fall  in  the  horizontal 
plane.  The  lateral  deviations  are  practically  the  same  whether  measured  on  the 
vertical  plane  perpendicular  to  the  line  of  fire,  or  on  the  horizontal  plane,  provided 
the  error  of  the  shot  in  range  be  not  too  great.  The  deviations  in  range,  however, 
differ  very  greatly  from  the  vertical  deviations,  the  ratio  between  them  being  the 
cotangent  of  the  angle  of  fall. 

425.  Since  the  mean  deviation  in  range,  jx,  is  related  to  the  mean  vertical  devia- 
tion, yy,  by  the  formula  y3-=yyCotM,  and  since  the  angle  of  fall  increases  and  its 
cotangent  correspondingly  decreases  with  increase  of  range  at  about  the  same  rate  as 
the  mean  vertical  deviation,  it  will  be  seen  that  the  mean  deviation  in  range  remains 
nearly  the  same  for  widely  different  ranges.  Thus,  for  example,  while  the  estimated 
mean  vertical  deviation  of  the  12"  gun  of  2800  f.  s.  initial  velocity  increases  from 
2.5  feet  at  1000  yards  range  to  10.5  feet  at  4000  yards  range,  the  corresponding 


254  EXTERIOR  BALLISTICS 

deviation  in  range  only  changes  from  119  yards  to  104  yards;  and  while  in  the  case 
of  smaller  guns  the  mean  deviation  in  range  decreases  more  rapidly,  still  the  change 
is  always  very  much  less  proportionately  than  the  change  in  range  itself. 

426.  The  principal  use  of  knowledge  of  the  mean  deviation  in  range  is  in  the 
regulation  of  gun  fire  by  observation  of  the  points  of  fall.  Suppose  the  axis  of  Z, 
in  Figure  44,  represents  the  water-line  of  the  target,  the  axis  of  Z  being  the  hori- 
zontal trace  of  the  vertical  plane  of  the  mean  trajectory,  and  let  the  distance  from  the 
axis  of  Z  to  the  dotted  lines  aa,  a' a',  hh,  h'b',  cc,  c'c',  dd,  d'd',  etc.,  represent  the  mean 
deviation  in  range,  y^,  of  the  gun.  Then  if  the  point  of  impact  be  on  the  axis  of  Z, 
that  is,  on  the  water-line,  half  of  all  the  shot  will  fall  short ;  if  it  be  on  a' a'  the  percent- 

X 


J 


a \--^ a 

c l-i c 

b \.Jl i, 

^ yi- 

a' ^--y ^' 

,' i.fi „' 

C \-ft c' 

d! ^_Zl a' 


Figure  44. 

age  of  shot  that  will  fall  short  will  be  increased  by  the  number  which  fall  between  the 
axis  of  Z  and  ala! ,  or,  from  the  table  of  probabilities,  it  will  be  50  H ^  =  79  per  cent. 

If  the  mean  range  be  still  further  short,  so  that  the  mean  point  of  impact  falls  on  h'h' , 

SS  9 
the  percentage  of  shorts  will  be  50  -f  —^ — \-  94  per  cent;  and,  finally,  if  the  mean  point 

of  impact  be  three  or  more  times  the  mean  deviation  short,  then  practically  all  the 
shot  will  fall  short.  The  same  reasoning  shows  that  if  no  shot  strike  short  of  the 
axis  of  Z,  the  mean  point  of  impact  is  three  or  more  times  the  mean  deviation  in 
range  beyond  the  axis  of  Z ;  if  about  6  per  cent  strike  short,  the  mean  point  of  impact 
is  about  twice  the  mean  deviation  in  range  beyond  the  axis  of  Z ;  and  if  about  21  per 
cent  are  short,  it  is  about  the  mean  deviation  in  range  beyond  the  axis  of  Z.  Thus, 
by  observation  of  the  percentage  of  shot  which  strike  short  it  is  possible  to  determine 
with  some  degree  of  accuracy  how  much  the  setting  of  the  sight  in  range  should  be 
increased  or  decreased  to  bring  the  mean  point  of  imj^aet  on  the  target. 


ACCURACY  AND  PROBABILITY  OF  GUN  FIRE 


255 


427.  Let  us  now  suppose  that  we  are  going  to  fire  salvos  from  a  battery  of  12" 
o-uns,  for  which  7  =  2900  f.  s.,  w  =  S70  pounds  and  c  =  0.61.  Let  us  also  assume  that 
we  have  a  vertical  target  30  feet  high  and  wide  enough  to  eliminate  the  necessity  for 
considering  lateral  deviations  due  to  accidental  errors.  Let  us  take  the  mean  errors 
of  the  gun  in  range,  first  as  40  yards,  next  as  GO  yards,  and  then  again  as  80  yards; 
and  also  that  they  are  approximately  the  same  at  all  ranges.  Let  us  also  assume  that 
the  mean  point  of  impact  is  at  the  center  of  the  water-line  of  the  target,  in  which 
case,  as  we  have  already  seen,  50  per  cent  of  the  shot  will  fall  short.  Let  us  also 
assume  that  the  three  mean  deviations  correspond  to  total  deviations  of  150,  200 
and  300  yards,  respectively.  Now  let  us  see  what  percentage  of  the  shot  in  each  salvo 
will  probably  hit,  at  a  range  of  7000  yards,  at  which  range  the  danger  space  for  a 
target  3U  feet  high  is,  by  the  range  table,  180  yards. 


■'l^- 


JO 


^ 


I    I 


Figure  45. 


1.  Mean  dispersion  in  range  40  yards;  maximum  dispersion  in  range  150  yards 
(Figure  45), 

,  30   _   1 

^^''"^  540 -18 


450      18 


y  =  25  feet 


That  is,  for  a  maximum  dispersion  of  150  yards,  or  450  feet,  no  shot  would  pass 
more  than  25  feet  above  the  water-line  of  the  target,  and  all  shots  that  do  not  fall 
short  would  hit.  Therefore,  by  our  assumption,  we  would  have  50  per  cent  of  shorts 
and  50  per  cent  of  hits. 


3oy 


Figure  46. 


2.  ^lean  dispersion  in  range  of  60  yards;  maximum  dispersion  in  range  of  200 
yards  (Figure  46). 


600       18 


y  =  33^  feet 


256 


EXTEKIOR  BALLISTICS 


Also  the  mean  dispersion  in  range  is  180  feet,  therefore  the  mean  vertical  dis- 
persion is 

y,  =  180tanto=  1^=10  feet 
'  18 

Our  problem  therefore  becomes  to  find  how  many  shot  will  pass  between  the  top  of 
the  target  and  a  line  parallel  to  it  and  3^  feet  above  it,  knowing  that  50  per  cent  of 
the  shot  fired  will  fall  short,  and  the  other  50  per  cent  will  pass  between  the  water- 
line  of  the  target  and  a  horizontal  line  33^  feet  above  it.  As  we  have  already  taken 
out  the  50  per  cent  of  the  shot  that  fall  short,  the  ^  disappears  from  the  formula, 
and  we  have 

Pj,(33)-Pj,(30)=.009 
33 


y 

7 


=  ^=3.3         Pj,(33)=.992 


10 
30 
10 


=  3.0 


Py{30)=.d83 
.009 


That  is,  .9  of  1  per  cent  of  the  shot  that  do  not  fall  short  will  pass  over  the  top  of  the 
target,  leaving  99.1  per  cent  of  them  as  hits.  Therefore,  of  the  100  per  cent  of  shot 
fired,  50  per  cent  will  fall  short;  99.1  per  cent  of  50,  or  49  per  cent  of  them,  will  hit; 
and  1  per  cent  of  them  all  will  go  over. 

3.  Mean  dispersion  in  range  80  yards ;  maximum  dispersion  in  range  300  yards. 


900       18^' 


50  feet 


yy- 


■240  x^^ 


40 


feet 


and  in  the  same  manner  as  in  2,  to  find  the  number  of  shot  that  will  pass  between 
the  top  of  the  target  and  a  horizontal  line  50  feet  above  the  water-line,  we  have 

Pj,(50)-Pj,(30)=.035 
fli  _  50x3 

y 
y 


40 

30x3 

40 


=  3.75 


=  2.25 


Pj,(50)=.9975 
P3,(30)=.9275 


.07 


Therefore  7  per  cent  will  go  over,  and  93  per  cent  will  hit  out  of  the  50  per  cent  that 
do  not  fall  short.  Therefore  we  have  that  there  will  be  50  per  cent  of  shorts,  46  per 
cent  of  hits  and  4  per  cent  of  overs.  Proceeding  similarly  for  other  ranges,  we  can 
make  up  a  table  like  the  following : 


Mean  point  of  impact  at  water-line. 

Mean  dispersion  in  range  of — 

Danger 

Range. 
Yds. 

40  yards 

60  yards. 

80  yards. 

space. 
Yds. 

Percentage  of — 

Percentage  of — 

Percentage  of — 

Shorts. 

Hits. 

Overs. 

Shorts. 

Hits. 

Overs. 

Shorts. 

Hits. 

Overs. 

7000 

50 

50 

0 

50 

49 

1 

50 

46 

4 

180 

10000 

nO 

49 

1 

50 

42 

8 

50 

35 

15 

108 

13000 

50 

42 

8 

50 

32 

18 

50 

25 

25 

70 

15000 

50 

36 

14 

50 

27 

23 

50 

21 

29 

55 

18000 

50 

28 

22 

50 

20 

30 

50 

15 

35 

39 

ACCTJEACY  AND  PEOBABILITY  OF  GUN  FIRE 


257 


The  above  mean  dispersions  are  less  than  have  been  experienced  at  recent  target 
practices. 

The  above  chances  of  hitting  are  based  only  on  vertical  errors ;  if  the  target  be 
short  they  will  be  materially  reduced  by  the  lateral  errors. 

428.  For  the  above  problem  let  us  now  suppose  that  the  mean  point  of  impact 
had  been  at  the  center  of  the  danger  space,  instead  of  at  the  water-line,  and  we  had 
desired  to  tabulate  the  same  data  as  before.  Let  us  start  with  the  range  of  7000 
yards,  for  which  the  danger  space  is  180  yards,  and  compute  the  results  for  a  mean 
dispersion  in  range  of  40  yards,  corresponding  to  a  total  dispersion  of  150  yards. 


A  /^   Mea.Tr.   Po/rxr    0f  Jmpctcf 

Figure  47. 


All  shot  that  fall  less  than  270  feet  short  of  the  mean  point  of  impact  are  hits. 
For  space  between  mean  point  of  impact  and  target, 

A  =  |I^  =  A:^2.25         P=.9275 

Therefore,  for  those  short  of  the  mean  point  of  impact,  92.75  per  cent  will  hit  and 
7.25  per  cent  will  fall  short.  But  only  50  per  cent  of  the  total  number  of  shot  fired 
fall  shot  of  the  mean  point  of  impact,  therefore  the  above  percentages  become,  of  the 
total, 

Hits    46.375  per  cent 

Shorts    3.625  per  cent 

For  the  space  between  A  and  C,  which  is  270  feet,  we  know  that  any  shot  that  falls 
between  A  and  C  hits,  and  any  that  falls  beyond  C  is  over.  Therefore  the  work  is  the 
same  as  the  above,  and  we  have 

Hits    46.375  per  cent 

Overs    3.625  per  cent 

Therefore  the  total  is 

Shorts    3.625  per  cent 

Hits    92.750  per  cent 

Overs    3.625  per  cent 


17 


258 


EXTERIOE  BALLISTICS 


Working  out  similar  data  for  the  other  ranges  and  dispersions  gives  us  the  following 
table : 


Mean  point  of  impact  at  center  of  danger  space. 

Mean  dispersion  in  range  of — 

Range. 

Danger 

Yds. 

40  yards. 

60  yards. 

80  yards. 

space. 
Yds. 

Percentage  of — 

Percentage  of — 

Percentage  of — 

Shorts. 

Hits. 

Overs, 

Shorts. 

Hits, 

Overs, 

Shorts. 

Hits. 

Overs. 

7000 

4 

93 

3 

12 

77 

11 

19 

63 

18 

180 

10000 

14 

72 

14 

24 

53 

23 

30 

40 

30 

108 

13000 

25 

49 

26 

32 

36 

32 

37 

27 

36 

70 

15000 

29 

42 

29 

36 

29 

35 

40 

21 

39 

55 

18000 

35 

31 

34 

40 

21 

39 

42 

16 

42 

39 

429.  From  the  tables  given  in  the  two  preceding  paragraphs,  if  we  assume  the 
mean  point  of  impact  on  the  target,  we  see  that,  as  the  mean  dispersion  increases, 
the  percentage  of  hits  decreases  very  rapidly. 

430.  It  will  also  be  seen  that,  to  get  the  greatest  possible  number  of  hits,  a 
greater  percentage  of  shorts  is  necessary  at  long  ranges  than  at  short  ranges. 

431.  It  will  also  be  seen  that,  where  the  mean  point  of  impact  in  range  is  at 
some  distance  from  the  target,  an  increase  in  dispersion  gives  an  increase  in  the 
number  of  hits,  which  is  in  accord  with  the  principles  previously  enunciated.  It  may 
be  shown  mathematically  that  the  mean  dispersion  for  maximum  efficiency  is  equal  to 
80  per  cent  of  the  distance  from  the  mean  point  of  impact  in  range  to  the  center  of 
the  danger  space. 

432.  From  what  has  been  said,  it  will  readily  be  seen  that,  in  controlling  the 
firing  of  salvos  from  a  battery  of  similar  guns,  we  desire  to  keep  a  certain  proportion 
of  the  shot  striking  short  of  the  target  in  order  to  get  the  maximum  number  of  hits. 
There  are  also  other  good  reasons  for  so  keeping  a  number  of  the  shot  striking  short. 
From  what  we  have  seen,  we  may  determine  certain  general  rules  which  will  govern 
the  spotter  in  thus  controlling  salvo  firing.  This  question,  however,  is  one  that  may 
more  appropriately  be  considered  at  length  in  the  study  of  another  branch  of 
gunnery,  so  there  will  be  no  further  discussion  of  it  in  this  book. 

433.  It  will  be  interesting  to  compare  the  results  of  actual  firing  with  the  com- 
puted results  in  some  one  case,  to  see  how  closely  the  two  agree,  and  to  get  some  idea 
of  the  correctness  for  service  purposes  of  percentages  determined  mathematically. 
Such  results  are  given  in  the  following  table  taken  from  Helie's  well-known  Traite 
de  Balistique.  It  represents  the  results  of  about  500  shots  fired  at  Gavre  from  a 
16.5-centimeter  rifle  at  various  ang-les  of  elevation : 


Probability  that  the  lateral  deviations  will 
not  exceed — 

7z 
4    ■ 

72 

2    ■ 

73- 

27«. 

37.. 

By  table 

Bv  firinc 

0.158 
0.176 

0.310 
0.300 

0.575 
0.592 

0.889 
0.885 

0.9R3 
0.988 

ACCURACY  AXD  PEOBABILITY  OF  GUN  FIEE  259 

EXAMPLES. 

1.  Supposing  a  row  of  gun  ports,  each  4  feet  high  by  6  feet  wide,  are  spaced 
18  feet  between  centers;  show  by  comparing  the  percentages  of  shot  which  would 
enter  a  port  that  it  woukl  be  better  to  aim  a  gun  whose  mean  vertical  and  lateral 
errors  are  each  5  feet  at  the  center  of  a  port  than  half  way  between  two  ports. 

Answe7\     9.5  per  cent  to  7.0  per  cent. 

2.  Compare  the  percentages  of  hits  on  ports  in  the  two  cases  of  Example  1  if 
the  mean  errors  of  the  gun  were  7.5  feet  instead  of  5  feet. 

Ansiver.     5.8  per  cent  to  5.8  per  cent. 

3.  The  12"  guns  of  a  certain  ship  made  69  per  cent  of  hits  on  a  target  15  feet  high 
by  20  feet  wide  at  1700  yards  range.  Supposing  the  mean  vertical  and  lateral  devia- 
tions to  have  been  equal,  what  was  their  approximate  value?  Supposing  through 
ignorance  of  the  range  the  sights  had  been  set  for  1870  yards,  thus  raising  the  mean 
point  of  impact  9  feet,  what  would  the  percentage  of  hits  have  been? 

Anstvers.     5  feet;  35.7  per  cent. 

4.  The  mean  errors,  laterally  and  in  range,  of  a  rifled  mortar  are  3.5  yards  and 
53  yards,  respectively,  at  a  mean  range  of  3357  yards.  What  is  the  chance  of  hitting 
a  ship's  deck  (taking  its  equivalent  area  to  be  a  rectangle  300  feet  by  60  feet)  when 
she  is  end  on;  (1)  if  the  mean  point  of  impact  be  at  one  corner  of  the  rectangle; 
(2)  if  it  be  at  the  center  of  the  rectangle ?  Answers.     .217 ;  .555. 

5.  What  are  the  chances  of  hitting  under  the  same  circumstances  as  in  Example 
4,  excepting  that  the  ship  is  broadside  on?  Answers     ..059;  .120. 

6.  At  a  range  at  which  the  mean  vertical  error  of  the  guns  equals  the  freeboard 
of  the  enemy,  what  is  the  ratio  of  the  respective  probabilities  of  hitting  her  when 
you  aim  at  her  water-line  and  when  you  aim  at  her  middle  height,  suppose  the  fire 
to  be  accurately  regulated  in  each  case  ?  What  is  the  same  ratio  at  a  range  at  which 
the  mean  vertical  error  is  only  half  the  freeboard  ? 

Ansivers.     .288  to  .310;  .445  to  .575. 

7.  A  torpedo-boat  steaming  directly  for  a  ship  at  24  knots  is  discovered  and 
fire  is  opened  on  her  at  1500  yards  range.  If  the  probability  of  a  single  3"  shot 
striking  her  is  .02,  and  there  are  eight  3"  guns  each  firing  12  rounds  a  minute  at  her, 
what  is  the  chance  that  she  will  be  struck  at  least  once  before  she  is  within  500  yards? 
What  is  the  chance  of  her  being  hit  at  least  twice?  Answers.     .911 ;  .694. 

8.  At  4000  yards,  a  ship  with  30  feet  freeboard  gives  a  danger  space  of  90  yards 
for  the  3"  gun  (F  =  2800  f.  s.)  and  the  mean  error  in  range  of  the  same  gun  at  4000 
yards  is  30  yards  (estimated).  How  closely  must  the  range  of  such  a  ship  be  known 
to  make  the  probable  percentage  of  hits  as  great  as  0.5  per  cent,  supposing  the  guns 
to  be  pointed  at  the  middle  of  the  freeboard?  Answer.     141  yards. 

9.  The  mean  error  in  range  of  the  12"  gun  (F  =  2800  f,  s.)  at  4000  yards 
range  is  100  yards  (estimated),  and  its  danger  space  for  30  feet  freeboard  is  300 
yards.  How  closely  must  the  range  be  known  to  make  the  probable  percentage  of 
hits  as  great  as  0.5  per  cent,  supposing  the  guns  to  be  pointed  at  the  middle  height 
of  the  freeboard?  Answer.     470  yards. 


260  EXTERIOE  BALLISTICS 

10.  Fire  is  opened  with  eight  3"  guns  on  a  torpedo-boat  coming  head  on  when 
she  is  at  1500  yards  range.  She  covers  100  yards  every  7.5  seconds,  and  each  gun 
fires  once  every  7.5  seconds.  The  mean  lateral  and  vertical  deviations  are  each  6  feet, 
and  the  target  offered  is  6  feet  high  by  15  feet  wide.  If  an  error  of  100  yards  in  the 
sight  setting  displaces  the  mean  point  of  impact  3  feet  vertically,  and  the  sights  are 
all  set  for  1000  yards  range,  what  is  the  probable  number  of  hits  while  the  boat 
advances  to  500  yards  range?  Aiiswer.     10  +  . 

11.  The  turrets  of  a  monitor  steaming  obliquely  to  the  line  of  fire  present  a 
vertical  target  consisting  of  two  rectangles,  each  24  feet  wide  by  12  feet  high,  and 
36  feet  from  center  to  center.  If  the  mean  errors  of  a  gun  be  12  yards  laterally 
and  8  yards  vertically,  would  it  be  better  to  aim  at  a  turret  or  half  way  between  them  ? 

Answers.     1st  case,  P=.057;  2d  case,  F=:.0G1. 

12.  A  gun  has  30  shell,  one  of  which,  if  landed  in  a  certain  gun  position,  would 
silence  the  gun  contained  therein.  The  gun  pit  is  10  yards  in  diameter,  and  the 
probability  of  hitting  it  with  the  gun  in  question  is  .05.  What  would  be  the  prob- 
ability of  silencing  the  gun,  using  all  the  ammunition?  Ansiver.     P  =  .7S5. 

13.  The  12"  guns  of  a  ship  made  68  per  cent  of  hits  on  a  target  15  feet  high  by 
20  feet  wide  at  1700  yards  range.  What  was  the  probable  value  of  the  mean  devia- 
tions, vertical  and  lateral  ?  Supposing  the  mean  deviation  to  be  proportional  to  the 
range,  what  percentage  of  hits  would  the  same  guns  make  on  the  same  target  at  3400 
and  at  5100  yards?  A^isivers.     5  feet;  25.9  per  cent;  12.6  per  cent. 

14.  If  the  probability  of  hitting  a  target  with  a  single  shot  is  .05,  what  will  be 
the  probability  of  making  at  least  two  hits  with  50  shots?  Answer.     .721. 

15.  What  is  the  greatest  value  of  the  mean  deviation  of  a  gun  consistent  with  a 
probability  equal  to  .90  of  its  making  at  least  one  hit  in  a  hundred  shots  on  a  gun 
port  2  feet  wide  by  4  feet  high?  Ansiver.     5.95  feet. 

16.  Compute  the  data  for  10,000  yards  contained  in  paragraph  427. 

17.  Compute  the  data  for  13,000  yards  contained  in  paragraph  427. 

18.  Compute  the  data  for  15,000  yards  contained  in  paragraph  427. 

19.  Compute  the  data  for  18,000  yards  contained  in  paragraph  427. 

20.  Compute  the  data  for  10,000  yards  contained  in  paragraph  428. 

21.  Compute  the  data  for  13,000  yards  contained  in  paragraph  428. 

22.  Compute  the  data  for  15,000  yards  contained  in  paragraph  428. 

23.  Compute  the  data  for  18,000  yards  contained  in  paragraph  428. 


APPENDIX  A. 

FORMS  TO  BE  EMPLOYED  IN  THE  SOLUTION  OF 

;rHE  PRINCIPAL  EXAMPLES  GIVEN  IN 

THIS  TEXT  BOOK. 

NOTES. 

1.  In  preparing  these  forms  the  problem  taken  has  been  the  8"  gim  (gun  K  in 
the  tables)  for  which  7  =  2750  f.  s.,  m;  =  260  pounds,  c  =  0.61,  generally  for  a  range 
of  19,000  A-ards.    More  specific  data  is  given  at  the  head  of  each  form, 

2.  In  the  problems  under  standard  conditions,  which  should  give  the  exact 
results  contained  in  the  range  tables,  it  should  be  borne  in  mind  that  the  latter  are 
given,  for  the  angle  of  departure  to  the  nearest  tenth  of  a  minute,  for  the  angle  of 
fall  to  the  nearest  minute,  for  the  time  of  flight  to  the  nearest  hundredth  of  a  second, 
etc.,  only.  Also  that  results  given  in  the  range  tables  are  entered  after  the  results 
of  the  computations  have  been  plotted  as  a  curve,  and  the  faired  results  are  those 
contained  in  the  tables.  Small  discrepancies  between  the  computed  results  and  those 
given  in  the  tables  may  therefore  sometimes  be  expected. 

3.  Also,  results  obtained  by  direct  computation  are  of  course  more  accurate 
than  those  obtained  by  taking  multiples  of  quantities  given  in  the  range  tables,  and 
small  discrepancies  may  be  expected  in  some  such  cases  between  computed  results 
and  those  taken  from  the  range  tables. 


Form 
No. 

Chapter  Example 
No.            No. 

1 

8                7 

INDEX  TO  FORMS  IN"  APPENDIX  A. 

Nature  of  Example. 
For  computing  0,  etc.,  for  a  given  range,  by  Ingalls'  method  of 
successive  approximations. 

2  8  8  For  computing  (p,   etc.,   for  a  given  range,  by   Ingalls'   method, 

knowing  /. 

3  8  9  For  computing  ((>,  etc.,  for  a  given  range,  by  Alger's  method  of 

successive  approximations. 

4  9  1  For  computing  the  elements  of  the  vertex  for  a  given  R  and  4>, 

by  successive  approximations. 

5  9  2  For  computing  the  elements  of  the  vertex  for  a  given  R  and  0, 

knowing  /. 

6  9  3  For  deriving  special  formulae  for  y  and  tan  6  for  a  given  tra- 

jectory. . 

7  9  4  For  computing  values  of  y  and  6  for  given  abscissae  in  a  given 

trajectory. 

8  10  1  For  computing  R,  etc.,  for  a  given  4>,  by  successive  approxima- 

tions. 

9  10  2  For  computing  R,  etc.,  for  a  given  <}>,  knowing  /. 

I      11  1  For  computing  0,  ^,  0,  t  and  v  for  an  elevated  or  depressed  target. 

11  12  2  For  computing  change  in  range  resulting  from  a  variation   in 

initial  velocity. 

12  12  3  For  computing  change  in  range  resulting  from  a  variation  in 

atmospheric  density. 

13  12  4  For  computing  change  in  range  resulting  from   a  variation  in 

weight  of  projectile;  method  by  direct  computation 

14  12  5  For  computing  change  in  range  resulting  from  a  variation  in 

weight  of  projectile;  method  by  using  Columns  10  and  12  of 
range  table. 

15  12  6  For  computing  change  in  position  of  point  of  impact  in  vertical 

plane  through  target  resulting  from  a  variation  in  the  setting 

of  the  sight  in  range. 
For  computing  the  drift  at  a  given  range. 
For  computing   sight-bar   height   and   set  of   sliding   leaf  for   a 

given  range  and  deflection. 
For  computing  the  effect  of  wind. 
For  computing  the  effect  of  motion  of  the  gun. 
For  computing  the  effect  of  motion  of  the  target. 
For  computing  the  penetration  of  armor. 
For  range  table  comnutations,  0,  etc. 
For  range  table  computations. 
For  range  table  computations. 

For  wind  and  speed  problems.     Real  wind. 

For  wind  and  speed  problems.     Apparent  wind. 
1  and  2  For  the  calibration  of  a  single  gun. 

For  the  calibration  of  a  ship's  battery. 


16 

13 

1 

17 

13 

2 

18 

14 

1 

19 

14 

2 

20 

14 

3 

21 

-  16 

— 

22 

16 

2 

23 

16 

3 

24 

16 

4 

25A'I 
25B  J 

17 

8 

26 

17 

9 

27 

18 

1 

28 

19 

1 

APPEN^DICES  263 


Form  No.  1. 

CHAPTER  S— EXAMPLE  7. 

FORM  FOR  THE  COMPUTATION  OF  THE  DATA  CONTAINED  IN  COLUMNS  2, 
3,  4  AND  5  OF  THE  RANGE  TABLES;  THAT  IS,  FOR  THE  VALUES  OF  THE 
ANGLE  OF  DEPARTURE  (4>),  ANGLE  OF  FALL  (o.),  TIME  OF  FLIGHT  (7") 
AND  STRIKING  VELOCITY  (v^,)  FOR  A  GIVEN  RANGE,  CORRECTING 
FOR  ALTITUDE  BY  A  SERIES  OF  SUCCESSIVE  APPROXIMATIONS,  THE 
ATMOSPHERE  BEING  CONSIDERED  AS  OF  STANDARD  DENSITY— 
INGALLS'  METHOD. 

FORMULAE. 
C,=K=  ^  ,Z=  ^;  sin  2cf>  =  AC;  Y-A"C  tan  (^;  loglog /=log  F  +  5.01765-10; 
tan  ti  =  B'  tan  <{>;  T  =  CT'  sec4>;  Vo]  =  Uo,  cos  <^  sec  w 

PEOBLEM. 

Cal.  =  8";  7  =  2750  f.  s.;  «'  =  260  pounds;  c  =  0.61;  Eange  =  19,000  yds.  =  57,000  ft. 

C,  =  K=  (from  Table  VI) colog  9.17654-10 

A'  =  57000    log  4.75587 

Zi  =  8558.75    log  3.93241 

^,  =  .08673+  -QQ^^^^y^-^^  -  -'''^l^''  -.084673   (from  Table  II) 

4i  =  .084673 log  8.92775-10 

.  C^=     log  0.8234G 

2<^i  =  34°   19'  36"    sin  9.75121-10 

01  =  17°   09'  48"  (first  approximation,  disregarding /) 

A  "->948-  ^  X  (--0064)  X71       50x0        .000273x71  .g^^.       .^  , ,     jj. 
A^  _^J48-  ^^^  X ^Q3^  +     ^^^     +        _QQ33^         _d0.7.5   (iable  11) 

^/'  =  3027.5    log  3.48109 

C^=     log  0.82346 

c/),  =  17°  09'  48"    tan  9.48974-10 


J\=     log  3.79429 

Constant    log  5.01765-10 


/i=     log  0.06485 loglog  8.81194-10 

C^=    log  0.82346 

Co=    log  0.88831 colog  9.11169-10 

X  =  57U00    ■ log  4.75587 

^2  =  7371.5    log  3.86756 

i,  =  .06520+^^0m^5  -  -00^^9^^X50^03333,3 

yl2  =  . 0638876    log  8.80542-10 

C,=    locr  0.88831 


202  =  29°  36'  14"    sin  9.69373-10 

02  =  14°  48'  07"  (second  approximation) 

A"     o.jQu       50    ,,  (-.0046)  x67    ,    50x0   ,    .001488x67      ^.^^^ 
A,  =2398-  ^0  X  -^^^ +  "loo-  +         .0035        =^^^^-^ 


264  APPEXDICES 

42"  =  2499.5    log  3.39786 

0^=    log  0.88831 

<f>,  =  U°  48'  07"    tan  9.42201-10 

Y.=    log  3.70818 

Constant   log  5.01765-10 

/,=     log  0.05319 loglog  8.72583-10 

C^=    log  0.82346 

C^=    log  0.87665 colog  9.12335-10 

Z  =  57000    ■ log  4.75587 

^3  =  7572.15    log  3.87922 


A3  =  .06851+  '^^^'^^^J^-^^  _  -.YY^f^-^^  =.067137 


.00170x72.15  _  .00520X50 
100  100 

43  =  .067137    log  8.82696-10 

C,=     log  0.87665 


2(/)3  =  30°  21'  22"    sin  9.70361-10 

(^3  =  15°   10'  41"  (third  approximation) 

A"     9i«-       50   ,,  (-.0048)  x68    ,    50x0^  .002237x68  _.;,^Q-,  ., 
As  =2460-  ^  X  ^0025 +  -Tor  +        ,0025        "^^^^-^ 

^3"  =  2591.1 .log  3.41349 

C^=    log  0.87665 

(^3  =  15°   10'  41"    tan  9.43342-10 

Y^=    log  3.72356 

Constant   log  5.01765-10 

f^=    log  0.05511 loglog  8.74121-10 

C^= log  0.82346 

C^=     log  0.87857 colog  9.12143-10 

X  =  57000 log  4.75587 

Z,  =  7538.75    log  3.87730 

A  -  06851+  'Q'^^QX^^-^^  -  •QQ^^.^.X^Q:^ .066569 
^*--"^°     ^  100  100 

^,  =  .066569 log  8.82327-10 

C^=    log  0.87857 

2<^,  =  30°   13'  10"    sin  9.70184-10 

^4  =  15°  06'  35"  (fourth  approximation) 

.„     ^,_       50   ,,  (-.0048)  X68   ,    50X0       .001669x68_ 
4.  =3465-—  X  ^0025 +  "lor  +         .0025        —575.6768 

A/'  =  2575.7    log  3.41090 

C^=    log  0.87857 

(^,'=15°  06'  35"    tan  9.43137-10 

Y^=    log  3.72084 

Constant    log  5.01765^10 

f^=    log  0.05476 loglog  8.73849-10 

C,=    log  0.82346  ■ 

^5=     log  0.87822 colog  9.12178-10 

Z  =  57000    log  4.75587 

^5  =  7544.85    log  3.87765 

A        n^e^i  L    .00170x44.85        .00520x50       nrrr-?^^ 
^,  =  .06851+.  ^^ ^^^^-~  =.0bbb,245 


APPEJ^DICES  2G5 

A,  =  MG672-i5    log  8.82395-10 

C.=    W  0.87822 


2<i>,  =  30°   14'  42"    sin  9.70217-10 

<f>,  =  lo°  07'  27"  (fifth  approximation) 
J  "_.2ifi=i       50   ^  ( -.0048)  X 68       50x0   ^  .001773x68      ...^  ^ 

^^  -^^^'-  Too  >< M25 +  -loT  +  — ^025—  =^^^^-^ 

^."  =  2578.5    log  3.41137 

C'5=     log  0.87822 

</>,  =  15°   07'  27"    tan  9.43175-10 

i\=    '. log  3.72134 

Constant    los  5.01765-10 


h=    log  0.05483 loglog  8.73899-10 

C^=    log  0.82346 

C^=    log  0.87829 colog  9.12171-10 

Z  =  57000    • loff  4.75587 


Ze  =  7543.65    log  3.87758 

i         n^Q-i   ,    .00170x43.65        .00520x50        n^rr-o 
/l,  =  .068ol+ — -^-^^—  =.0666o2 

^Q  =  .066652    log  8.82381-10 

C«=    loff  0.87829 


208  =  30°   14'  21"    sin  9.70210-10 

(/)g  =  15°  07'  10"  (sixth  approximation) 

i  "     oiaK       50   ^  (-.0048)  x 68   ,    50x0    .    .001752x68      ^.^^^ 

4e"  =  2577.9    .log  3.41126 

C^=     log  0.87829 

<^6  =  15°  07'  10"    tan  9.43166-10 

1\=    log  3.72121 

Constant   los  5.01765  - 10 


f,=    log  0.05481 loglog  8.73886-10 

C\=    log  0.82346 

C,=    log  0.87827 colog  9.12173-10 

X  =  57000    loff  4.75587 


Z,  =  7544.0    log  3.87760 

J        nrQ-1^ -00170  X44       .00520x50      nrrfi=;Q 
A,  =  M^ol  +  —^^ --^00 =  -0^66o8 

^,  =  .066658    log  8.82386-10 

C,=    log  0.87837 

2(^,  =  .10°   14'  30"    sin  9.70213-10 

<;().  =  15°  07'  15"  (seventh  approximation) 

J"     OAc-       50   ^  (-.0048)  X  68  ^  50x0  ^  .001758x68     „.^q  , 
^^  =^^^'-Wq''  :0025 +  -T00-  + :0025— =^^^^-^ 


266  APPENDICES 

A/'  =  2578.1    log  3.41130 

C,=   log  0.87827 

<j>l  =  15°  07'  15"    tan  9.43170-10 

Y,=    log  3.72127 

Constant log  5.01765  - 10 

/,=    log  0.05481 loglog  8.73892-10 

Ci=    log  0.82346 

Cs  = log  0.87827 

As  C^  =  Ct,  the  limit  of  accuracy  has  been  reached  and  the  work  of  approxima- 
tion can  be  carried  no  further. 

By  the  preceding  work  we  have  derived  the  following  data  for  the  remainder  of 
the  problem : 

</>  =  15°  07'  15"         Z  =  7544.0         log  (7  =  0.87827 

From  Table  II,  with  the  above  value  of  Z, 

log5'  =  .2652+  :^0^4  ^  .002^6^x50  ^.g,^^ 

r  =  4.600+  :^^  _  -1^3X50  ^^g,^, 

«^=1086-^^  +  ^^^  =1097.0 
100  100 

B'=    log  0.26751 

C=    log  0.87827 

(^  =  15°  07'  15"    tan  9.43170-10.  .sec  0.01530 cos  9.98470-10 

r'  =  4.554    log  0.65839 

Wa,  =  1097    log  3.04021 

(0  =  26°   34'  40"    tan  9.69921-10 sec  0.04850 


7  =  35.642    lo2f  1.55196 


t;„  =  1184.2    log  3.07341 

EESULTS. 

By  above  computations.  As  given  in  range  table. 

<^ 15°  07'  15"  15°  07'  00" 

oy 26°   34'  50"  26°   35'  00" 

T 35.642  seconds  35.64  seconds 

Vui 1184.2  foot-seconds  1184  foot-seconds 

Note  to  Form  No.  1. — The  number  of  approximations  necessary  to  secure  correct 
results  increases  with  the  range,  therefore  problems  for  shorter  ranges  will  not  involve 
so  much  labor  as  the  one  worked  out  on  this  form. 


APPENDICES  2Gr 


Form  No.  2. 

CHAPTEE  8— EXAMPLE  8. 

FORM  FOR  THE  COMPUTATION  OF  THE  VALUES  OF  THE  ANGLE  OF  DEPART- 
URE (0),  ANGLE  OF  FALL  (o.),  TIME  OF  FLIGHT  (T)  AND  STRIKING 
VELOCITY  (i^^)  FOR  A  GIVEN  RANGE,  MAXIMUM  ORDINATE  AND 
ATMOSPHERIC  CONDITION. 

FORMULA. 
C=  -^^^'  ^—~(T'>  sin2<^  =  AC;  tan  w  =  5' tan  ^ ;  T  =  CT'  sec  (f>;  r^^w^  cos  c^  sec  w 

PEOBLEM. 

Cal.  =  8";  F  =  2750  f.  s.;  w  =  2G0  pounds;  c  =  O.CA;  Eange  =  19,000  yards  =  57,000 
feet;  Barometer  =  28.33";  Thermometer  =  82.7°  F.;  Maximum  ordinate  =  5261 
feet. 
From  Table  III,  8  =  .9139G;  |F  =  3507',  hence  /=  1.0962  from  Table  V. 

K=  (from  Table  VI) log  0.82346 

/=1.0962    '..     log  0.03989 

8  =  .91396    log  9.96093-10.  .colog  0.03907 

C=    log  0.90242 colog  9.09758-10 

Z  =  57000    log  4.75587 


Z  =  7136.0    log  3.85345 

From  Table  XL        ^^^^^^^^_^  ^00158x36  _  .00475x50  ^^^^^^^^ 

log 5' =  .2551+  ^^^^  +  --^0^X50  ^_,.,,^ 

rp,     ,  OOP  ,   .089x36       .163x50  _,  ^qq^ 
T  =4.238+  -^^^ ^^^—  -4.1885 

110  1       10x36    ,    34x50      ..^r/A 


268  APPENDICES 

4  =  .060204 log  8.77963-10 

B'=    log  0.25677 

r  =  4.1885    log  0.62206 

w^=  1137.4   log  3.05591 

C=    W  0.90242 loir  0.90242 


2(^  =  28°  44' 38"    sin  9.68205-10 

</)  =  14°  22'  19" tan  9.40864-10.  .sec  0.01381.  .cos  9.98619-10 

a)  =  24°50'0S"    tan  9.66541-10 sec   0.04215 


T  =  34.537   los  1.53829 


i;a,  =  1214.1   log  3.08425 

RESULTS. 

<;!. 14°   22'  19". 

o) 24°   50'  08". 

T 34.537  seconds. 

t'w 1214.1  foot-seconds. 

Note  to  Form  No.  2. — To  solve  the  above  problem  with  strict  accuracy  the  maximum 
ordinate  should  not  be  used,  but  the  approximation  method  should  be  employed  as  in 

Form  No.  1,  starting  with  a  value  of  01  =  -^,  and  proceeding  as  shown  on  Form  No.  1. 

o 

In  order  to  get  a  series  of  shorter  problems  for  section  room  work,  an  approximately 

correct  value  of  the  maximum  ordinate  is  given  in  the  above  data,  from  which,  by  the  use 

of  the  value  of  /  obtained  from  Table  V,  an  approximately  correct  value  of  C  may  be 

determined    without   employing   the   longer   method    of   Form    No.    1.      The    results    are 

sufficiently  accurate  to  enable  the  process  to  be  used  for  the  purpose  of  instruction  in  the 

use  of  the  formulae  subsequently  employed. 


APPENDICES  2G9 


Form  No.  3. 

CHAPTER  8— EXAMPLE  9. 

FORM  FOR  THE  COMPUTATION  OF  THE  VALUES  OF  THE  ANGLE  OF  DEPART- 
URE (<^),  ANGLE  OF  FALL  (o.),  TIME  OF  FLIGHT  (T")  AND  STRIKING 
VELOCITY  ( v^)  FOR  A  GIVEN  RANGE,  CORRECTING  FOR  ALTITUDE  BY 
A  SERIES  OF  SUCCESSIVE  APPROXIMATIONS,  FOR  GIVEN  ATMOS- 
PHERIC CONDITIONS— ALGER'S  METHOD;  NOT  USING  TABLE  IL 

FORMULA. 

PROBLEM. 

Cal.  =  8";  7  =  2750  f.  s.;  «'  =  260  pounds;  c  =  0.61;  Range  =  19,000  yards  =  57,000 
feet;  Variation  in  7  due  to  wind  =-25  f.  s.;  Effective  initial  velocity  =  2725 
f.  s.;  Barometer  =  30.50";  Thermometer  =10°  F. 

w  =  260    log  2.41497 

8  =  1.144    log  0.05843 colog  9.94157-10 

c  =  0.61    log  9.78533-10.  .colog  0.21467 

£^2  =  64    log  1.80618 colog  8.19382-10 

Ci=    log  0.76503 colog  9.23497-10 

Z  =  57000    log  4.75587 

.^=^„^-5'p.  =  A*S\=   9791.2    log  3.99084 

8v=   2565.2  From  Table  I. 

S,   =12356.4     A„^  =  1673.19       r„^  =  7  650 
«<,  =  939.8  ^y=   lt)0--3         rF  =  0.819         7f  =  .04832 

AAi  =  1572.96       Ari  =  6.831 

A.4,  =  1573    log  3.19673  1, 

A»S,  =  9791.2   log  3.99084  jSubtractive. 

Mi  =  .16065    log  9.20589-10 

Ao^ 

/f  =  . 04832 

-=%-i-  -/f=  .11233    log  9.05050-10 

A^i 

Ari  =  6.831 log  0.83448 

•    C,=    log  0.76503 log  0.76503 

2(^1  =  40°   50'  16"    sin  9.81553-10 

</,,  =  20°   25'  08"    (first  approximation) sec  0.02819 

Ti=    log  1.62770 

r^^  =     log  3.25540 

g  =  32.2    log  1.50786 

8    log  0.90309 colog  9.09691-10 

7^  =  7247.1    log  3.86017 

|7i  =  4831.4,  hence  /i  =  1.1359,  from  Table  V 


270  APPENDICES 

/i  =  1.1359    log  0.05534 

C\=    log  0.76503 

0^=    log  0.82037 colog  9.17963-10 

^=57000    log  4.75587 

AS,=   8619.7   log  3.93550 

Sr=  2565.2  4^^  =  1269.96  r„^  =  6.443 

^11184.9  Ay=   100^  r;  =  0.819 

"0?  ■ 

u^=    1007.1  A.42  =  1169.73  AT^^  5.624 

A^o^  1169.7    log  3.06808 

ASo  =  861d.7    W  3.93550 


\  Subtractive. 


^^^  =  .13570    log  9.13258-10 

/f  =  . 04832 


aa  

^2  -/^=.08738    log  8.94141-10 

Aro  =  5.624    log  0.75005 

C\=    log  0.82037  log  0.82037 


20,  =  35°   17'  48"    sin  9.76178-10 

0o  =  17°  38'  54"   (second  approximation) sec  0.02094 

1^=    log  1.59136 

T/^    log  3.18272 

g  =  32.2   log  1.50786 

8    colog  9.09691  - 10 


F2  =  6130.4    log  3.78749 

§r2  =  4086.9,  hence  /,  =  1.1136 

/2  =  1.1136    log  0.04673 

C^=    log  0.76503 

C^=     log  0.81176 colog  9.18824-10 

Z=  57000    log  4.75587        ' 


a5'3=   8792.4    log  3.94411 

Sv=   2565.2  1„^  =  1323.79  r„^=6.617 

o„„=113o<.6  _  

u^=     996.05  AA3  =  1223.56  Ar3  =  5.798 


APPENDICES 

AAr,  =  1223.56    log  3.08764 

AS,  =  8792A    log  3.94411 


A/4 

^^  =  .13916    log  9.14353 

/r  =  . 04833 


r  Subtract!  ve. 


^  -Iy  =  .0d084: .log  8.95828-10 

Ar3  =  5.798    log  0.76328 

C,= log  0.81176 log  0.81176 

2«>3  =  36°  04'  46"    sin  8.77004-10 

<^3  =  18°  02'  23"    (third  approximation) sec  0.02190 

Ts=    log  1.59694 

Ts-=    log  3.19388 

g  =  32.2   log  1.50786 

8    colog  9.09691  - 10 

^3  =  6290.0    log  3.79865 

fF3  =  4193.3,  hence  /a^  1.1168 

/3  =  1.1168    log  0.04797 

Ci=    log  0.76503 

C\= log  0.81300  colog  9.18700-10 

Z  =  57000    log  4.75587 

AS^=   8767.4    log  3.94287 

Sv=   2565.2  4„^  =  1315.96  r„^=:6.592 

^„^  =  l"S^  ^.==J00^  Ty  =  0M9_ 

i{l=     997.6  A.44  =  1215.73  Ar,  =  5.773 

A.44  =  1215.73    log  3.08483 

A*S',  =  8767.4    log  3.94287 


f  Subtractive. 


4-^^i  =.13866    log  9.14196-10 

A04 

/f  =  . 04832 

-Mi-/   =.09034    log  8.95588-10 

Ar,  =  5.773    log  0.76140 

C^=    log  0.81300 log  0.81300 

2<^,  =  35°   58'  04"    sin  9.76888-10 

</>4  =  17°   59'  32"    (fourth  approximation) sec  0.02175 

T^=    log  1.59615 

T,^=    log  3.19230 

^  =  32.2    log  1.50786 

8    colog  9.09691  - 10 

F,  =  6267.2    log  3.79707 

§7^  =  4178.1,  hence /4  =  1.1163 


372  APPENDICES 

/4  =  1.1163    log  0.04778 

C^=    log  0.76503 

^5=     log  0.81281 colog  9.18719-10 

Z:=57000    log  4.75587 

^8,=   8771.2    log  3.94306 

^F=   2565.2  A«^=1316.97  T„^  =  6.596 

^„„  =  1133674  Ar=J00^3  Tf^O-819 

u^-=     997.4  AA5  =  1216.74  AT,  =  5.777 

A45  =  1216.7    log  3.08518  ^ 

A^,  =  8771.2    log  3.94306  jSubtractive. 

^^  =  .13871    log  9.14212-10 

/f  =  . 04832 
-^^-/^  =  .09039    log  8.95612-10 

Ar5  =  5.777    log  0.76170 

C^=    log  0.81281 log  0.81281 

2.^5  =  35°  58'  20"   sin  9.76893-10 

<^.  =  17°   59'  10"   (fifth  approximation) sec  0.02176 

T^=    log  1.59627 

T^-=    log  3.19254 

^  =  32.2    log  1.50786 

8    colog  9.09691-10 

75  =  6270.6    log  3.79731 

§75  =  4180.4,  hence  /3  =  1.1164 

/5  =  1.1164   log  0.04782 

Ci=    log  0.76503 

C^-    log  0.81285 colog  9.18715-10 

Z=  57000    W  4.75587 


A»S6=   8770.4    log  3.94302 

Sv=   2565.2  A„^  =  1316.97  r„^  =  6.596 

^„„=Ii^^6  AU^m^         r;=o^ 

wl=     997.4  A/l6  =  1316.74  AT8  =  5.777 


APPENDICES  273 

A.4«  =  1216.7    loff  3.08518 

A.4. 


a5«  =  8769.5    log  3.94302  rSubtractive. 


—=^=.13873    log  9.14216-10 

/r  =  . 04832 

M6._7^^.09041    log  8.95622-10 

A/bg 

Ar6  =  5.777    log  0.76170 

Cr=    loff  0.81285 log  0.81285 


2<^6  =  35°   59'  10"    sin  9.76907-10 

(^6  =  17°  59'  35"   (sixth  approximation) sec  0.02177 

Tg=    log  1.59632 

T^-=    log  3.19264 

g  =  32.2    log  1.50786 

8    colog  9.09691-10 


F6  =  6272.0    log  3.79741 

fF6  =  4181.3,  hence  /e^  1.1164 

/e  =  1.1164    log  0.04782 

Ci=    log  0.76503 

C,=    log  0.81285 

We  see  that  Ct  =  Cq.    Therefore  further  work  will  be  simply  a  repetition  of  the 
last  two  approximations,  and  the  limit  of  accuracy  has  been  reached. 


IS 


274  APPENDICES 

From   the   preceding  work,   therefore,   we   have   the   following    data   for   the 
remainder  of  the  problem : 

Wa,=997.4  ^=:.13873  Ar  =  5.777  log  (7  =  0.81285 

-=^=.13873  Z„,  =.31477  From  Table  I. 

77  =  . 04832  -^=.13873 

■|^-/.  =  .09041  7„„- ^=.17604 

-^ -/;.  =  . 09 041  log  8.95622-10 

/„^- ^  =  .17604  log  9.24561-10 

Ar=5.777    log  0.76170 

C=    log  0.81285 log  0.81285 log  0.81285 

2    colog  9.69897-10 

Wc.  =  997.45    W  2.99887 


2<^  =  35°  59'10"    ..sin  9.76907-10 
</>  =  17°  59' 35"    2  sec  0.04354 sec  0.02177.. cos   9.97823-l( 


(0  =  32°  18' 28"    tan  9.80097-10 sec  0.07303 


r=39.4745    log  1.59632 

i;„  =  1122.4 log  3.05015 

EESULTS. 

<^ 17°   59'  35". 

(o 32°   18'  28". 

T 39.4745  seconds. 

Vo) 1122.4  foot-seconds. 

Note  to  Form  No.  3. — The  number  of  approximations  necessary  to  secure  correct 
results  increases  with  the  range,  therefore  problems  for  shorter  ranges  will  not  involve  as 
much  labor  as  the  one  worked  out  on  this  form. 


APPENDICES  275 


Form  No.  4. 

CHAPTER  9— EXAMPLE  1. 

FORM  FOR  THE  COMPUTATION  OF  THE  ELEMENTS  OF  THE  VERTEX 
FOR  A  GIVEN  ANGLE  OF  DEPARTURE  (c/>)  AND  GIVEN  ATMOSPHERIC 
DENSITY,  CORRECTING  FOR  ALTITUDE  BY  A  SERIES  OF  SUCCESSIVE 
APPROXIMATIONS. 

FORMULA. 

loglog/  =  log  F  +  5.01T65  — 10;  Xo  =  Czq;  1^^  =  01^'  sec  <^;  1^0  =  ^0  cos  ^ 

PROBLEM. 

Cal.  =  S";   F  =  2750  f.  s.;  it;  =  260  pounds;  c=0.61;  Range  =  19,000  yards; 
</)  =  15°   07'  00";  Barometer  =  29.42";  Thermometer  =  75°  F. 

K-    log  0.82346 

§  =  .96344    loo-  9.98383-10.  .colog  0.01617 


(?,=    log  0.83963 colog  9.16037-10 

2c;!)  =  30°   14'  00"    sin  9.70202-10 

ao;  =  .0728435 log  8.86239-10 

A"     or-n       50    ^  (-.0053)  X  69    ,    50x0   ,    .0001435x69      „^oq  o 

^^^   ='*^'^-rOO  ""  .0028 +  ^00"  + ^028 =^^^^-^ 

A/'  =  2738.8    log  3.43756 

C\=    log  0.83963 

<^  =  15°   07'  00"    tan  9.43158-10 

Fi=    log  3.70877 

Constant    log  5.01765  -  10 


/i.=     log  0.05326 loglog  8.72642-10 

C\=     log  0.83963 

Co=    log  0.89289 colog  9.10711-10 

2(^  =  30°   14'  00"    sin  9.70202-10 

fl,;=.0644355    log  8.80913-10 

^  "_.iQQQ       50   ^  (-.0046)  X  67   ,50x0    ,    .0020355x67  _g-,,  g 
A,  -.398-  ^  X  ^^-g +  -3^  + ^^^^ -2014.2 

^o"  =  2514.2    ■ log  3.40040 

C'„=    log  0.89289 

(6=15°   07'  00"    tan  9.43158-10 


l\_=    log  3.72487 

Constant    log  5.01765-10 

/,=    log  0.05527 loglog  8.74252-10 

C'i=    log  0.83963 

C,=    log  0.89490 colog  9.10510-10 

2</>  =  30°   14'  00"    sin  9.70202-10 

00/  =  .0641385    .     log  8.80712  - 10 

^  "-o.^oQ       50    ^  (-.0046)  X  67   ,    50x0   ,    .0017385x67    ^..-^^  o 
^3  --Jb-—  X -^^^  ^-^OT"^         :0025  ^-^^^-^ 


276  APPENDICES 

A,"  =  2506.2 log  3.39901 

C,=     log  0.89490 

<;!>  =  15°   07'  00"    tan  9.43158-10 

Y,=     log  3.72549 

Constant    loo-  5.01765  - 10 


f.=    log  0.05535 loglog  8.74314-10 

C,=     10^0.83963 


C^=    log  0.89493 colog  9.10502-10 

2(^  =  30°   14'  00"    sin  9.70202-10 

f7o/=.06417    log  8.80704-10 

A  "-oqqs       50   ^(-.0046)x67   ,    50x0   ,    .0Q1727X  67  _o^^.  ^ 

A,  -^39«-^x     -^^^      +-ior+     :oo25~-^^"''-^ 

A4"=:2505.9    log  3.39896 

C^=    log  0.89498 

<;i,  =  15°   07'  00"    tan  9.43158-10 

r,  =  5315.25    log  3.72552 

Constant    log  5.01765  — 10 


f,=    log  0.05536 loglog  8.74317-10 

C,=     log  0.83963 

Cr=    log  0.89499 colog  9.10501-10 

2.^  =  30°   14'  00"    sin  9.70202-10 

aj=  .0641256    log  8.80703  - 10 

A"     o-^QQ       50   ,,  (-.0046)x67    ,    50x0       .0017256  X  67  _o^ 
A,  -2398-  ^^-  X  -^^ +  -j^  +  —^0025  -^..0o.9 

A^"  =  Ai",  therefore  the  limit  of  accuracy  has  been  reached,  and  we  have  for  the 
data  for  the  remainder  of  the  problem : 

a '  =  .0641256  102  0  =  0.89499 


,^  =  4100+  i^,  r.0017256-  50x_(^,0046l 


=  4261.0 


.0025  L  100 

,,     onnp  ,    .063X61        .079x50  _^^..,t, 
t,  =2.0o6+-j^^ ^-—— 2.0o49 

,.Q_      21x61    ,    66x50      ..-,_  0 
^0  =  1^90 ^^  +  -^^  =  1GU.2 

C=    log  0.89499 log  0.89499 

<;i.  =  15°   07'  00" sec  0.01529 cos  9.98471-10 

2o  =  4261    log  3.62951 

V  =  2.0349 log  0.30854 

M,  =  1615.2    log  3.20822 


a:,  =  33458    log  4.52450 


L  =  16.551    loir  1.21882 


i'o  =  1559.3 log  3.19293 

EESULTS. 

Xo 11152.7  yards. 

tj^  =  Y 5315.25  feet. 

t^ 16.551  seconds. 

r„ 1559.3  foot-seconds. 

Note  to  Form  No.  4. — The  number  of  approximations  necessary  to  secure  correct 
results  increases  with  the  range,  therefore  problems  for  shorter  ranges  will  not  involve 
as  much  labor  as  the  one  worked  out  on  this  form. 


APPENDICES  277 

Form  No.  5. 

CHAPTER  9— EXAMPLE  2. 

FORM  FOR  THE  COMPUTATION  OF  THE  ELEMENTS  OF  THE  VERTEX 
FOR  A  GIVEN  ANGLE  OF  DEPARTURE  AND  GIVEN  ATMOSPHERIC 
DENSITY,  GIVEN  ALSO  THE  MEAN  HEIGHT  OF  FLIGHT  FROM  WHICH 
TO  CORRECT  FOR  ALTITUDE. 

FOEMUL.E. 

C  =  ^  K;  %'  =A  =  ^^^^  ;  yo  =  Y  =  A"C  tan  <f> ;  Xo  =  Czo ;  t^  =  CtJ  sec  cf> ;  v^  =  u^  cos  <f> 

PEOBLEM. 

Cal.  =  8",;  F  =  2750  1  s.;  iv  =  260  pounds;  c  =  0.61;  Eange=:19,000  yards; 

</)  =  15°   07'  00';  Barometer  =  29.42";  Thermometer  =  75°  F. 

Mean  height  of  flight  =  3543  feet. 

71=    log  0.82346 

/=1.0973 log  0.04033 

8  =  .96344    loff  9. 98383-10.. colog  0.01G17 


C=   log  0.87996 colog  9.12004-10 

2(^  =  30°   14'  00"    sin  9.70202-10 

<  =  . 0663835    log  8.82206-10 

.//_9,fi.       50   ^   (-.0048)X68   ,    50x0        .0014835x68  _gy^  . 
^   -'^^^^"100^"~~:0025  +^00"   +  :0025^         -^^^U.b 

,^^4200+ -i^  [.0014835- ^QX(-JQ^^)]  =4355.3 

V^2.099+  -064 X^55.3_  .082^X50^^0^3^ 

,^^,      20x55.3   ,    66x50      .^^^  ^ 

''-  =  ^^"^ 100- + -Too- =  ^^^^-^ 

C=    log  0.87996 log  0.87996.  .log  0.87996 

yl"  =  2570.6    log  3.41003 

,^  =  15°  07' 00"    .tan  9.43158-10 sec  0.01529.  .cos  9.98471-10 

Zo  =  4355.3    log  3.63902 

V  =  2.0934    log  0.32085 

Wn  =  1595.9    log  3.20300 


y  =  5267.1    los  3.72157 


,  =  33035    loff  4.51898 


^0  =  16.447    log  1.21610 

i;o  =  1540.7    log  3.18771 

EESULTS. 

3-0 11011.7  yards.  ^o 16.447  seconds. 

1/0  =  Y 5267.1  feet.  v^ 1540.7  foot-seconds. 

Note  to  Form  No.  5. — To  solve  the  above  problem  with  strict  accuracy,  the  mean 
height  of  flight  should  not  be  used,  but  the  approximation  method  employed  in  Form 

No.  4  should  be  followed,  starting  with  a  value  of  Ci  =   ,.  ,  and  proceeding  as  given  on 

o 
Form  No.  4.  In  order  to  get  a  series  of  shorter  problems  for  section  room  work,  an 
approximately  correct  value  of  the  mean  height  of  flight  is  given  in  the  above  data  (and 
in  Example  2),  from  which,  by  the  use  of  the  value  of  /  obtained  from  Table  V,  an 
approximately  correct  value  of  C  may  be  determined  without  employing  the  longer  method 
of  Form  No.  4.  The  results  are  sufficiently  accurate  to  enable  the  process  given  in  this 
form  to  be  used  for  purposes  of  instruction  in  the  use  of  the  formulte  subsequently 
employed. 


278  APPE^sTDICES 


Form  No.  6. 

CHAPTEB  9— EXAMPLE  3. 

FORM  FOR  THE  DERIVATION  OF  THE  SPECIAL  EQUATIONS  FOR  COMPUTING 
THE  VALUES  OF  THE  ORDINATE  AND  OF  THE  ANGLE  OF  INCLINATION 
OF  THE  CURVE  TO  THE  HORIZONTAL  AT  ANY  POINT  OF  THE  TRA- 
JECTORY WHOSE  ABSCISSA  IS  KNOWN,  WITH  ATMOSPHERIC  CON- 
DITIONS STANDARD ;  CORRECTING  FOR  ALTITUDE. 

FORMULA. 

J  _  sin  2<^  .  tan  </j  .  .        .        ±      n     tan  6  .  -        ,s 

^-      Q      ;  ^~~A      (^■^~^)*'^'  tan^  =  ^-^  (A-a') 

PPtOBLEM. 

CaL  =  S";  7  =  2750  f.  s.;  w  =  260  pounds;  c=0.61;  Eange  =  19,000  yards; 

<A  =  15°  07'  00";  logC  =  0.87827  (value  corrected" for  /  from 

work  in  example  on  Form  No.  1). 

C=    colog  9.12173-10 

20  =  30°  14'  00"    sin  9.70202-10 


A  =  .066643 log  8.82375-10 colog  1.17625 

<f>  =  lo°  07'  00"    tan  9.43158-10 

^-^=4.0535    log  0.60783 

RESULTS. 

?/ =  4.0535  (.066643- a)  .-c 
tan  ^  =  4.0535(.066643-a') 

Note  to  Form  No.  6. — To  determine  the  above  equations  with  accuracy  for  any  given 
trajectory  in  air,  the  value  of  log  C  must  be  determined  by  the  process  of  approximation 
given  on  Form  No.  1,  for  the  range  for  which  the  special  equations  are  desired.  This 
value  of  log  C  must  then  be  used  as  was  done  in  the  above  problem.  An  approximate 
result  may  be  obtained  by  determining  the  value  of  /  by  means  of  the  maximum  ordinate 
given  in  the  range  table,  from  which  the  value  of  E  may  be  approximately  corrected  for 
altitude. 


APPENDICES  279 


Form  No.  7. 

CHAPTER  9— EXAMPLE  4. 

FORM  FOR  THE  COMPUTATION,  FOR  ANY  GIVEN  TRAJECTORY,  OF  THE 
ABSCISSA  AND  ORDINATE  OF  THE  VERTEX  AND  OF  THE  ORDINATE  AND 
OF  THE  ANGLE  OF  INCLINATION  OF  THE  CURVE  TO  THE  HORIZONTAL 
AT  ANY  POINT  OF  THE  TRAJECTORY  WHOSE  ABSCISSA  IS  KNOWN, 
HAVING  GIVEN  THE  SPECIAL  EQUATIONS  FOR  /  AND  TAN  6  FOR  THE 
GIVEN  TRAJECTORY. 

FOEMUL.^. 

y,  =  Y  =  A"Ctan<f>;  x,  =  Cz,;  y=  ^   (A-a)x;  tan  ^=  *^  (A-a') 

PEOBLEM. 

Cal.  =  8";  7  =  2750  f.  s.;  w  =  260  pounds;  c  =  0.61;  Eange  =  19,000  yards  =  57,000 

feet;  log  (7  =  0.87827;  «/)  =  15°   07'  00";  ?/  =  4.0535(.0G6643-a)a:; 

tan  ^  =  4.0535 (.066643 -a')  ;  a-^^SOOO  yards  =  24,000  feet; 

a;2=:  16,000  yards  =  42,000  feet. 

For  Vertex:     From  data,  /I  =  .066643,  and  for  vertex  ao=A,  therefore,  from 
Table  II  for  ao'  = -066643. 

A"     oiPK      50   ,,  ( -.0048)  X 68   ,    50x0    ,    .001743  X  68  _o^^^  ^ 
^   =^^*^^- Too  ^  .0025 +  -T00~  +        .0025        '^^^^'^ 


Zo  =  4200  + 


001713      (--0018)  X50 
■^^^^^'^     ■  100 


=  4365.7 


.0025 

A"  =  2577.7    log  3.41123 

(7=    log  0.87827 log  0.87827 

<^  =  15°   07'  00"    tan  9.43158-10 

2n  =  4365.7    log  3.64005 


a;o  =  32,985  feet=  10,995  yards log  4.51832  

^„  =  F  =  5261.1  feet   log  3.72108 

For  x^  =  8000  yards  =  24,000  feet : 

C=    colog  9.12173-10 

a;i  =  24000    log  4.38021 

2  =  3176.4    log  3.50194 

.,^^,   ,    .00075x76.4       .00131x50       ni779Q 
a=.01.81+ ^^ j^3^^—  =.017728 

,.^.0,08+. -0019X76.4  _  :003 1^x50  ^_,^,,02 

A  =  .066643       A  =  .066643 
a=. 017728       a' =  .040702 


A-a=.048915  log  S. 68945-10 

yi-a'  =  . 025941 log  8.41399-10 

^^^  =4.0535    log  0.60783 log  0.60783 

A  O  G 

a:i  =  24000    log  4.38021 

2/1  =  4758.7    log  3.67749 


'=6°  02'  10"    tan  9.02182-10 


280  APPENDICES 

For  jr.  =  16,000  yards  =  48,000  feet : 

C=    colog  9.12173-10 

a;,  =  48000 log  4.68134 

2  =  6352.8    log  3.80297 


^   ^^^^^  ,  .00137X52.8 
a  =  .05030+     ^^^ 

_. 00386  X  50  ^Q^9Qg3 

,   iQ^Q,  .0044x52.8 
a  =.1358+    ^^^    - 

.0105X50  ^.^3,g.3 

^  =  .066643       A  = 

.066643 

a=. 049093       a'  = 

.132873 

A-a=.017550    log  8.24428-10 

A-a'=(-).066230    (-)log  8.82105-10 

^^^  =4.0535    log  0.60783 log  0.60783 

a;,  =  48000    loir  4.68124 


?/o  =  3414.7    log  3.53335 

^2=(-)15°  or  39"    (-)taii  9.42888-10 

For  point  of  fall,  a:  =  Z=  19,000  yards  =  57,000  feet: 

C=    colog  9.12173-10 

a;=  57000   log  4.75587 

2  =  7544.0 log  3.87760 

a=  .06851+  -Q^Yo^^^  -  ^^^^^^  =  .066658 
a'=  .1946+  :^255X44  _  .0140  x50^^^,,,,q 

A=         .066643         A=         .066643 
a=         .066658         a'=         .190020 


l-a=(-).000015    (-)log  5.17609-10 

A -a'=(-). 123377    ". (-)log  9.09125-10 

^~^  =4.0535    log  0.60783 .' . .  .log  0.60783 

a;=57000    W  4.75587 


?/„=(-)3.4657  feet   (-)log  0.53979 


^„=(-)26°   34'  15"    (-)tan  9.69908-10 

RESULTS. 
For  vertex.  For  a?i=:8000  yards.  For  a-;Z=  16,000  yards.  For  point  of  fall. 

0:0  =  10,995  yards  7/^  =  4758.7  feet    ^,  =  3414.7  feet  ?/u,=  (- )3.4657  feet 

?/o=F=  5261.1  feet        ^i  =  6°02'10"       ^2=  (  -  )15°  01' 39"  ^a,=  (- )26°  34' 15" 

Note  to  Form  No.  7. — In  the  above  problem  of  course  the  ordinate  at  the  point  of  fall 
should  be  zero.  The  angle  6  at  the  point  of  fall  should  equal  — w;  for  which  the  work 
gives  6=  ( — )26°  34'  15",  and  the  range  table  gives  w  =  26°  35'  00".  These  comparisons 
give  an  idea  of  the  degree  of  accuracy  of  the  above  method. 


APPENDICES  281 

Form  No.  S. 

CHAPTER  10— EXAMPLE  1. 

FORM  FOR  THE  COMPUTATION  OF  THE  VALUES  OF  THE  RANGE  (R),  ANGLE 
OF  FALL  (io),  TIME  OF  FLIGHT  (T)  AND  STRIKING  VELOCITY  M  FOR 
A  GIVEN  ANGLE  OF  DEPARTURE  (</>)  AND  ATMOSPHERIC  CONDITION, 
CORRECTING  FOR  ALTITUDE  BY  A  SERIES  OF  SUCCESSIVE  APPROXI- 
MATIONS. 

FOEMUL^. 

C=  -t  K;  ao'  =  /l  =  ^HL^  ;  X  =  CZ;  Y=A"C  tan  cf> ;  \og\og  f  =  \og  r  + 5.01765 -10; 

tan  (x)  =  B'  tan  cf>;  T  =  C T'  sec  <l> ;  v^  =  Wc^  cos  <^  sec  w 

PROBLEM. 

Cal.  =  S";  7  =  2750  f.  s.;  w  =  260  pounds;  c  =  0.61;  <^  =  15°   07'  00"; 
Barometer  =29.00";  Thermometer  =  82°  F. 

K=    log  0.82346 

S  =  .937    loir  9.97174-10.  .coloff  0.02826 


C\=    log  0.85172 colog  9.14828-10 

2<i  =  30°   14'  00"    sin  9.70202-10 


ao/=. 0708435    log  8.85030-10 

A  >'-orm       50   ^  (-.0051)  X 69    ,    50x0    ,    .0008435 X 69  _..Q^;, 
A,   -.601-  ^  X  -^^ +  -j^  +  -^^ -2687.7 

4/'  =  2687.7    log  3.42938 

C^=    log  0.85172 

d>  =  15°  07'  00"    tan  9.43158-10 


1\=    log  3.71268 

Constant    locv  5.01765-10 


/,=    log  0.05374 loglog  8.73033-10 

C\=    log  0.85172 

C„=     log  0.90546 colog  9.09454-10 

2</)  =  30°   14'  00"    sin  0.70202-10 

Cfo.;  =  . 0625985    log  8.79656-10 

J  "_.oQQQ       50   ^  (-.0046)  X 67   ,    50X0    ,    JX)01985x67_„,..^ 
A,   -.3J8-  —  X  -^^  +  ~j^  +  -^^  -24bD.O 

A."  =  2465  log  3.39183 

C\_=   log  0.90546 

6  =  15°   07'  00"    tan  9.43158-10 


F„=    log  3.72886 

Constant   log  5.01765-10 


/.,  =    log  0.05578 loglog  8.74651  - 10 

C\=     log  0.85172 

C,  =    log  0.90750 colog  9.09250  - 10 

2(h  =  30°   14'  00"    sin  9.70202-10 


ff   '  =  .062304    log  8.79452-10 


'03 


A  "-033i_  i^  X  (-•QQ-^5)x67       50^  +  -00^304X67  ^^^^g 
^3  --J-^i      -^QQ  ^  ^0024  ^    100  .0024 


282  APPENDICES 

A^"  =  24:58.1    log  3.390G0 

Cs=    log  0.90750 

<f>  =  15°  07'  00"    tan  9.43158-10 

^3=    log  3.72968 

Constant   log  5.01765-10 

/3=     log  0.05589 loglog  8.74733-10 

Ci=    log  0.85172 

C^=    log  0.90761 colog  9.09239-10 

24>  =  30°  14'  00"    sin  9.70202-10 


ao/=.0622885    log  8.79441-10 

A  "_oQQi        50   ^  (-.0045)  X67    ,    50x0    ,    .0022885x67  _„.-^  ^ 

"^^  —^"^1-  100  ^  ^024 +  ^^r  +  :0024         — ^^^-^ 

.4/'  =  2457.7    log  3.39053 

C^=    log  0.90761 

<f>  =  15°  07'  00"    tan  9.43158-10 

1\=    log  3.72972 

Constant   log  5.01765-10 


/,=    log  0.05589 loglog  8.74737-10 

Ci=    log  0.85172 

C,=    log  0.90761 

which  equals  log  C^ ;  therefore  the  limit  of  approximation  has  been  reached,  and  we 
have  the  following  data  : 

A  =  .0622885  log  (7  =  0.90761  <^  =  15°   07'  00" 

■     ^  Z  =  7100+  -^[.002785-  50x  (-JM75)j  ^^,g^  ^ 

log5'  =  .2578+  :0025X67^  ^  -0017x50  ^,,^3, 

rr'_A  Q9.V  I   .090X67.9       .165x50_,  ^^>^ 

^  -^'^^^^  — 100 — ioo~  -^-30^^ 

,,,,       10x67.9    ,    33x50_^^9O7 

C=    log  0.90761 log  0.90761 

Z  =  7267.9    log  3.86141 

<f>  =  15°  07'  00"    tan  9.43158-10.  .sec  0.01529.  .cos  9.98471-10 

B'=    log  0,26035 

r  =  4.3056    log  0.63403 

w„  =  1123.7    log  3.05065 


Z  =  58752    loff  4.76902 


w  =  26°  11' 44"    tan  9.69193-10 sec  0.04707 

r  =  36.052    log  1.55693 

i'^=  1208.9    log  3.08243 

EESULTS. 

R 19,584  yards.  T .'. 36.052  seconds. 

to 26°   ir  44".  Vo; 1208.9  foot-seconds. 

Note  to  Fokm  No.  8. — The  number  of  approximations  necessary  to  secure  correct 
results  increases  with  the  angle  of  departure,  therefore  problems  for  a  smaller  angle  of 
departure  will  not  involve  so  much  labor  as  the  one  worked  out  on  the  form. 


APPENDICES  283 

Form  No.  9.  CHAPTER  10— EXAMPLE  2. 

FORM  FOR  THE  COMPUTATION  OF  THE  VALUES  OF  THE  HORIZONTAL 
RANGE  (R),  ANGLE  OF  FALL  (o.),  TIME  OF  FLIGHT  (T)  AND  STRIKING 
VELOCITY  (i'^)  FOR  A  GIVEN  ANGLE  OF  DEPARTURE  (c/>),  ATMOS- 
PHERIC CONDITION  AND  MAXIMUM  ORDINATE. 

FORMULA. 

C=^K;  A  =  ^^^^^  ;   X=CZ;   tan  w  =  5' tan  «^ ;    T  =  CT'sec<f>; 
0  C 

Va,  =  Uco  COS  <f>  sec  0) 

PROBLEM. 

Cal.  =  8";  7  =  2750  f.  s.;  w  =  260  pounds;  c=0.61;  <^  =  15°  07'  00";  Barometer 

=  29.00";  Thermometer  =  82°  F.;  Maximum  ordinate  =  5400  feet; 

fF  =  3G00  feet. 

R=    log  0.82346 

/=  1.099    log  0.04100 

8  =  .937    log  9.9717-i-lO.  .colog  0.02826 


C=    log  0.89272 colog  9.10728-10 

2<^  =  30°   14'  00"    sin  9.70202-10 

A  =  .064461    log  8.80930-10 


^=''''+wk 


100 


1       P'      oron  I   -0024x5    ,    .0023x50       ^..^^ 
log  B  =  .2620  +  -3^^^  +        100       =  .26407 

r  =  4.508  +  ^^^1^  -  ''^^^^  ^^  =4.4276 

,,.1095-^5  +  ^0=1110.1 

C=    log  0.89272 log  0.89273 

Z  =  7405    log  3.86953 

(^  =  15°   07'  00"    tan  9.43158-10.  .sec  0.01529.. cos  9.98471-10 

n'=     log  0.26407 

r  =  4.4276    log  0.64617 

2^^  =  1110.1    log  3.04536 


Z  =  57843    log  4.76225 

(0  =  26°  23'  24"    tan  9.69565-10 sec  0.04780 


r  =  35.824    los  1.55418 


ra,  =  1196.4    log  3.07787 

RESULTS. 

From  work  with 
From  above  work.  same  data  on 

Form  No.  8. 

R 19,281  yards         19,584  yards 

oj 26°  23'  24"  26°  11'  44" 

T 35.824  seconds  36.052  seconds 

Vco 1196.4  foot-seconds  1208.9  foot-seconds 

Note  to  Form  No.  9. — To  solve  the  above  problem  with  strict  accuracy  it  must  be  done 
as  shown  on  Form  No.  8.  In  order  to  get  a  series  of  shorter  problems  for  section  room 
work,  an  approximately  correct  value  of  the  maximum  ordinate  is  given  and  employed  as 
above.  The  comparison  of  results  by  the  two  methods  given  at  the  bottom  of  the  above 
work  gives  an  idea  of  the  degree  of  inaccuracy  resulting  from  the  employment  of  the 
method  given  on  this  form. 


281  APPEJ^DICES 


Form  No.  lOA. 

CHAPTER  11— EXAMPLE  1  (WHEN  y  IS  POSITIVE). 

FORM  FOR  THE  COMPUTATION  OF  THE  VALUES  OF  THE  ANGLE  OF  ELEVA- 
TION {xp)  (THE  JUMP  BEING  CONSIDERED  AS  ZERO),  ANGLE  OF  INCLI- 
NATION TO  THE  HORIZONTAL  AT  THE  POINT  OF  IMPACT  {6),  AND  TIME 
OF  FLIGHT  TO  (0  AND  REMAINING  VELOCITY  AT  (i' )  THE  POINT  OF 
IMPACT  WHEN  FIRING  AT  A  TARGET  AT  A  KNOWN  HORIZONTAL  DIS- 
TANCE FROM  THE  GUN  AND  AT  A  KNOWN  VERTICAL  DISTANCE  ABOVE 
THE  HORIZONTAL  PLANE  OF  THE  GUN,  FOR  GIVEN  ATMOSPHERIC 
CONDITIONS. 

EOEMUL^. 

C=^K;  tan  p  =  ~;  z  =  ^;  sin  2(^x  =  aC ;  shi(2(f)  —  p)  =sin  p(l  +  cot  p  sin  2(f)x)  ; 
A=  — -^p^  ;  tan  6=        ^  (A—a');  t  =  Ct'  sec  cp;  v  =  ucoscj>  sec  6;  \l/  =  cf>  —  p 

PEOBLEM. 

Cal.  =  8";  F  =  27o0  f.  s.;  w  =  260  pounds;  c  =  0.61;  Gun  below  target  900  feet; 
Horizontal  distance  =  18,000  yards  =  54,000  feet;  Maximum  ordinate  =  4170 
feet;  Barometer  =  29.00";  Thermometer  =  40°  F.;  §7  =  2980  feet. 

K=    .' log  0.82346 

/=1.0804    log  0.03358 

8=1.021    W  0.00903 colour  9.99097-10 


C=    log  0.84801 colog  9.15199-10 

y  =  900 log  2.95424 Subtractive. 

a;=54000 loff  4.73239 log  4.73239 


p  =  0°  57'  18"    tan  8.22185-10 


2  =  7662.6    log  3.88438 

n-  A^noi,    -00172X62.6        .00532x50  _  nrqro^ 
a_.070.1+ ^ — -.068627 

a'=.2001+  -^'056X62.6  _  .0143^x50  ^  -^^^^g 
^:.1077-^^  + ^^=1086.5 
i' ^4.692  + ^09^^  -  :i:?^=  4.6627 


APPEXDICES  285 

C=     log  0.84801 

a  =  .068627    log  8.83649-10 

2cf>^=    sin  9.68450-10 

L  p  =  0°   57'   18"    cot  1.77815 

cot  77  sin  2(/)^.  =  29.016    log  1.46265 

1  +  cot/;  sin  2</)^  =  30.016    log  1.47735 

p=   0°   57'  18" sin  8.22185-10 

24>-p  =  30°  01'  02"    sin  9.69920-10 

p=   0°  57'  18" 

2.^  =  30°  58'  20"    sin  9.71149-10 

<^  =  15°  29'   10" 

/;=   0°   57'  18" 

^^  =  14°   31'  52" 

C=     colog  9.15199-10 

^  =  .07303    log  8.86348-10 

A=         .07303 
a=         .19646 


^i_fl=(_). 12343    (-)log  9.09143-10 

<;!>  =  15°  29'10"    tan  9.44258-10.. sec  0.01606.  .cos   9.98394-10 

A  =  .07303    log  8.86348-10.  .colog  1.13652 

r  =  4.6627 ^ log  0.66864 

«  =  1086.5 log   3.03603 

C=    log  0.84801 


^=(-)25°  05'  38"    (-)tan  9.67053-10 sec   0.04306 

^  =  34.097 log  1.53271 


t'  =  1156.2    log   3.06303 

The  range  table  gives  for  7^  =  18,500  yards (/)  =  14°   26.9' 

for  7?  =  18,600  yards 0  =  14°   34.9' 

Therefore,  for  an  angle  of  elevation  of  1/^  =  14°   31.9',  the  sight  setting  in  range 

would  be  2^=18500+  ^2S^  =18562.5  yards. 

o 

EESULTS. 

i}, 14°  31'  52". 

6 (-)25°  05'  38". 

t 34.097  seconds. 


V 


.1156.2  foot-seconds. 


Setting  of  sight  in  range.  . .  .18,550  yards. 

Note  to  Form  No.  lOA. — Note  that  the  work  on  this  form,  for  a  target  higher  than  the 
gun,  is  the  same  as  that  on  Form  No.  lOB  for  the  same  problem  with  the  gun  higher  than 
the  target,  down  to  and  including  the  determination  of  the  value  of  cot  p  sin  20j.,  except 
that  the  sign  of  that  quantity  and  of  the  position  angle  (p)  is  positive  in  Form  No.  lOA, 
and  negative  in  Form  No.  lOB.  Compare  the  results  obtained  on  these  two  forms,  having 
in  mind  the  remarks  made  in  paragraphs  191,  192  and  193  of  Chapter  11  of  the  text. 


286  APPENDICES 


Form  No.  lOB. 

CHAPTER  11— EXAMPLE  1  (WHEN  y  IS  NEGATIVE). 

FOEM  FOR  THE  COMPUTATION  OF  THE  VALUES  OF  THE  ANGLE  OF  ELEVA- 
TION (.//)  (THE  JUMP  BEING  CONSIDERED  AS  ZERO),  ANGLE  OF  INCLI- 
NATION TO  THE  HORIZONTAL  AT  THE  POINT  OF  IMPACT  {6),  AND 
THE  TIME  OF  FLIGHT  TO  {i)  AND  REMAINING  VELOCITY  AT  (i^)  THE 
POINT  OF  IMPACT  WHEN  FIRING  AT  A  TARGET  AT  A  KNOWN  HORI- 
ZONTAL DISTANCE  FROM  THE  GUN  AND  AT  A  KNOWN  VERTICAL  DIS- 
TANCE BELOW  THE  HORIZONTAL  PLANE  OF  THE  GUN,  FOR  GIVEN 
ATMOSPHERIC  CONDITIONS. 

FOEMULJ^. 
C=  '^K;  tan  p  —  ^;  z=-^  \  sin  2(f>j:  =  aC;  sin(2(f>  —  p)  =sin  p(l  +  coip  sin  2(^j)  ; 
^_  sm_^  .  ^^^  ^_    an  (f>  ^j^_^'^  .  f  —  Qf  gee  <^;  v  =  u  cos  ^  sec  6;  ij/  =  (j>  —  p 

PEOBLEM. 

CaL  =  8";   7  =  2750  f.  s.;  iv  =  260  pounds;  c  =  0.61;  Gun  above  target  900  feet; 
.     Horizontal   distance  =  18,000   yards  =  54,000   feet;    Maximum   ordinate  =  4470 
feet;  Barometer =29.00";  Thermometer  =  40 °  E.;  fF  =  2980  feet. 

E=    log  0.82346 

/=  1.0804    log  0.03358 

S  =  1.021    W  0.00903.. coloff  9.99097-10 


C=    log  0.84801 colog  9.15199-10 

rr  =  54000    log  4.73239J log  4.73239 

p={-)0°  57'  18"...(-)tan  8.22185-10 


2  =  7662.6    log  3.38438 

.      n-noi  ,    .00172x62.6       .00532x50       ^^0^0^ 
a=.0.021+ — j^^^— =.068627 

„>      ooni  J   -0056x62.6       .0143x50       -.nrir 

^=-^^^  +  — 100 locf-^-^^^^^ 

^=1077--^^^|-  +^^^:^1086.5 

^  =  4.692+^^93^^6  _  -175  X  50  ^^  ^^^^ 


APPENDICES  287 

C=    log  0.84801 

a  =  . 068627    W  8.83649-10 


2<j>jc=    sin  9.68450-10 

p={-)Q°   57'  18"    (-)cot  1.77815 

cot/jsm2<^^=(-)29.016   (-)log  1.46265 

l  +  cot/Jsm20.r=(-)28.O16 (-)log  1.44741 

;;=(-)0°   57'  18"    (-)sin  8.22185-10 

2<l>-p=         27°   50'  10"    (  +  )sin  9.66926-10 

p={-)   0°   57'  18" 

2<t>=         26°  52'  52"    sin  9.65528-10 

(f>=         13°  26'  26" 
I  p=(-)   0°  57'  18" 

"  iA=  14°  23'  44" 

C=    coloff  9.15199-10 


A  =  .06416    log  8.80727-10 

A=         .06416 
a'=         .19646 


A-rt'=  (-).13230    (-)log  9.12156-10 

cf>  =  13°  26'  26"    tan  9.37836-10.  .sec  0.01206.  .cos   9.98794-10 

A  =  .06416    .log  8.80726- 10.. colog  1.19273 

f  =  4.6627    log  0.66864 

w  =  1086.5    log  3.03603 

C= log  0.84801 


^zr(-)26°  13' 58"   (-)tan  9.69265-10 sec  0.04720 

^  =  33.784    W  1.52871 


t;  =  1178.0 log  3.07117 

The  range  table  gives  for  i2  =  18,400  yards (^  =  14°  19.0' 

for  i2  =  18,500  yards <^  =  14°  26.9' 

Therefore,  for  an  angle  of  elevation  of  ;/'  =  14°  23.7',  the  sight  setting  m  range 

would  be  E  =  18400+  '^''^^^^^  =18459.5  yards. 

EESULTS. 

ip 14°  23'  44". 

(I (-)26°   13'  58". 

t 33.784  seconds. 

V 1178.0  foot-seconds. 

Setting  of  sight  in  range.  ..  .18,450  yards. 

Note  to  Form  No.  lOB. — Note  that  the  work  on  this  form,  for  a  target  lower  than  the 
gun,  is  the  same  as  that  on  Form  No.  lOA  for  the  same  problem  with  the  target  higher 
than  the  gun,  down  to  and  including  the  determination  of  the  value  of  cot  p  sin  2(;6x, 
except  that  the  sign  of  that  quantity  and  of  the  angle  of  position  (j))  is  minus  in  Form 
No.  lOB  and  plus  in  Form  No.  lOA.  Compare  the  results  obtained  in  the  two  cases,  having 
in  mind  the  remarks  made  in  paragraphs  191,  192  and  193  of  Chapter  11  of  the  text. 


288  APPENDICES 


Form  No.  11. 

CHAPTER  12— EXAMPLE  2. 

FORM  FOR  THE  COMPUTATION  OF  THE  CHANGE  IN  RANGE  RESULTING 
FROM  A  VARIATION  FROM  STANDARD  IN  THE  INITIAL  VELOCITY, 
OTHER  CONDITIONS  BEING  STANDARD. 

FORMULA. 

PROBLEM. 

Case  1. — Correcting  for  Altitude  by  Table  V. 

Cal.  =  8";  7  =  2750;  w  =  260  pounds;  c  =  0.61;  Range  =  19,000  yards  =  57,000  feet; 
Maximum  ordinate  =  5261  feet;  Variation  from  standard  of  7= +75  f.  s.; 
§7=3507  feet. 

K=    log  0.82346 

/=1.0962    log  0.03989 

C=    log  0.86335 colog  9.13665-10 

Z  =  57000   log  4.75587 

Z  =  7807.6 log  3.89252 

A           nn^^c^r  ,    -00012X7.6        .00049x50        nnf;Q9di 
Af4  =  .00556-^ — ^j^ =.0053241 

„      ^o^^  ,   .0040x7.6        .0093x50        ,000^ 

^=-^^^^+     100 lor— =-^^^^^ 

Afa  =  .0053241    log  7.72625-10 

7^  =  19000    log  4.27875 

5  =  .13335    log  9.12500-10 colog  0.87500 

87=(  +  )75  log  1.87506 

A7=100 io?  2.00000 coloff  8.00000-10 


AEf=(  +  )568.93  yards log  2.75506 

PROBLEM. 

Case  2. — Using  Corrected  Value  of  0  Obtained  by  Successive  Approximations  on 

Form  Xo.  1. 

Cal.  =  8";  7  =  2750  f.  s.;  w  =  260  pounds;  c  =  0.61;  Range  =  19,000  yards  =  57,000 

feet;  log  (7  =  0.87827 ;  Variation  from  standard  of  7= +75  f.  s. 

C=    colog  9.12173-10 

Z  =  57000    log  4.75587 


Z  =  7544.0    log  3.87760 

A,, ^.00520+  :00012X44  _  .00046^X50  ^.0050228 

£  =  .1261+  -^^^^  -  -003^8^X50^,^33^ 


APPENDICES  289 

Ar^  =  .0050328    : log  7.7009-i-lO 

7?  =  19000    log  4.27875 

^=.12337    log  9.09121-10 colog  0.90879 

8V=(  + )  75   log  1.8750G 

A7=100   log  2.00000 colos  8.00000-10 


Ai2F=(  +  )580.15 log  2.76354 

Note  to  Form  No.  11. — The  method  of  Case  2  is  of  course  the  more  accurate,  and  gives 
the  range  table  result.  The  method  shown  in  Case  1  is  introduced  to  give  practice  in  the 
use  of  this  formula  without  the  necessity  of  taking  up  the  successive  approximation 
method  in  order  to  determine  the  value  of  C  accurately. 


19 


290  APPENDICES 


Form  No.  12. 

CHAPTEE  12— EXAMPLE  3. 

FORM  FOR  THE  COMPUTATION  OF  THE  CHANGE  IN  RANGE  RESULTING 
FROM  A  VARIATION  FROM  STANDARD  IN  THE  DENSITY  OF  THE  ATMOS- 
PHERE, OTHER  CONDITIONS  BEING  STANDARD. 

FOEMUL^. 
C-   fw   .  y_  X         -p  _      (B-A)R  ^  AO 

PEOBLEM. 

Case  1. — Correcting  for  Altitude  by  Table  V. 

Cal.  =  8";  7=2750  f.  s.;  w  =  260  pounds;  c  =  0.61;  Eange  =  19,000  yards  =  57,000 

feet;  Maximum  ordinate  =  5261  feet;  Variation  in  density  =+15 

per  cent;  fF  =  3507  feet. 

K=    ' log  0.82346 

/=  1.0962    log  0.03989 

C=    log  0.86335 colog  9.13665-10 

Z  =  57000   log  4.75587 

Z  =  7807.6   log  3.89252 

A  =  .07368+  -QQ^^^Q^^-^  -  :00^<50  =.071035 
5^.1377+^^^^^  _  -009^3^X50  ^133354 

^  =  .133354 
4  =  .071035 


S-A  =  .062319    log  8.79462-10 

i2  =  19000    log  4.27875 

^=.13335    log  9.12500-10 colog  0.87500 

\n 

~-  =  .15    , log  9.17609  - 10 

Ai2fi=  ( - )  1331.9    log  3.12446 

PEOBLEM. 

Case  2. — Using  Corrected  Value  of  C  Obtained  by  Successive  Approximations  on 

Form  No.  1. 

Cal.  =  8";  7  =  2750  f.  s.;  w  =  260  pounds;  c=0.61;  Eange  =  19,000  yards  =  57,000 

feet;  log  C  =  0. 87827;  Variation  in  density  =+15  per  cent. 

C=    colog  9.12173-10 

Z  =  57000 loff  4.75587 


Z  =  7544.0    log  3.87760 

A       AfiQr^i,    -001^0X44       .00520x50       ^acRKQ 
A  =  .06851+— j^^ 100—  =-^^^^^^ 

R      19^1   ,    .0038x44       .0088x50       ,.,0...., 
£  =  .1261+  —^^ ^^^^^  =.Uooa 


APPENDICES  291 


5=. 123373 

^  =  .066658 


5-4  =  .056714    log  8.75369-10 

i?=  19000    log  4.27875 

5=. 12337    log  9.09121-10 colog  0.90879 

4^  =  .15    log  9.17609  - 10 

ARs=  ( -  )  1310.1    log  3.11732 

Note  to  Form  No.  12. — The  method  of  Case  2  is  of  course  the  more  accurate,  and  gives 
the  range  table  result.  The  method  shown  in  Case  1  is  introduced  to  give  practice  in  the 
use  of  this  formula  without  the  necessity  for  taking  up  the  successive  approximation 
method  in  order  to  determine  the  value  of  C  accurately. 


292  APPENDICES 

Form  No.  13. 

CHAPTER  12— EXAMPLE  4. 

FORM  FOR  THE  COMPUTATION  OF  THE  CHANGE  IN  RANGE  RESULTING 
FROM  A  VARIATION  FROM  STANDARD  IN  THE  WEIGHT  OF  THE  PRO- 
JECTILE, OTHER  CONDITIONS  BEING  STANDARD.  DIRECT  METHOD 
WITHOUT  USING  COLUMNS  10  AND  12  OF  THE  RANGE  TABLES. 

FORMULA. 

C=  i^  ;  87=  -0.36  —  V; 

cd-  w 

Ai?,  =  ifi,'  +  AK,"=^-4  X  4^  XB+  (^=^  X  ^ 
^^  ^  *^         B  AK  B  w 

PROBLEM. 
Case  1. — Correcting  for  Altitude  by  Table  V. 

Cal.  =  8";  7  =  2750  f.  s.;  m;  =  260  pounds;  c  =  0.61;  Range  =  19,000  yards  =  57,000 
feet;  Maximum  ordinate  =  5261  feet;  Variation  in  weight=  +10  pounds; 

§7  =  3507  feet. 

K=    log  0.82346 

/=1.0962   log  0.03989 

C=    log  0.86335 colog  9.13665-10 

Z  =  57000   log  4.75587 

Z  =  7807.6    log  3.89253 

A..  =  .00556+:«M|X1,6  _  .00049x50^.^^^3,^^, 

^  =  .07368+ -Q^^^X^-^  --00556X50  ^^^^^3^ 

■        B  =  .1377+-Q7^f-^  -  ^^^^^0  =.133354 

Aw;=  +10   log  1.00000 

w  =  260   colog  7.58503-10 

7  =  2750   log  3.43933 

.36   log  9.55630-10 

8V={-)    log  1.58066 

Afa  =  . 0053241   log  7.72625-10 

72  =  19000    log  4.27875 

5=. 13335    log  9.12500-10 colog  0.87500 

S7=(-)    (-)log  1.58066 

A7'  =  100    log  2.00000 colog  8.00000-10 

AB«,'=(-)288.84    log  2.46066 

5  =  . 133354 

^  =  .071035 


5-A  =  .062319    log  8.79462-10 

22  =  19000    log  4.27875 

£  =  .13335 colog  0.87500 

Aw=  +10   log  1.00000 

w  =  260  colog  7.58503-10 

AE«;"=  (  +  ) 341.51    log  2.53340 

ARw=    +     52.67  yards, 


APPENDICES  293 

hence  an  increase  in  weight  gives  an  increase  in  range  for  this  gun  at  this  range, 
therefore  this  quantity  would  carry  a  negative  sign  in  Column  11  of  the  range  table 
for  this  range. 

PROBLEM. 

Case  2. — Using  Corrected  Value  of  C  Obtained  by  Successive  Approximations  on 

Form  No.  1. 

Cal.  =  8";  7  =  2750  f.  s.;  w  =  260  pounds;  c  =  0.61;  Eange  =  19,000  yards  =  57,000 

feet;  log  (7  =  0.87827;  Variation  in  weight  =  +10  pounds. 

C=    colog  9.12173  - 10 

Z  =  57000   log  4.75587 

Z  =  7544.0    log  3.87760 

A,,  =  .00520  +  ^^^^^^  -  :000i^0  ^  ^^.^Q^.g 
A  =  .06851  +  :00mx44  _  -00520^X50  ^  ..g^^g 

P      ,^^,^.0038x44       .0088x50       -.ooory. 
5=.1261  +  — ^^^— ^^^— =.123o72 

Aw=  +10   log  1.00000 

tt;  =  260   colog  7.58503-10 

7  =  2750  log  3.43933 

.36 log  9.55630-10 

hV=  (  -  )    log  1.58066 

Afa  =  .0050228    log  7.70094-10 

E  =  19000    log  4.27875 

£  =  .12337    log  9.09121-10 colog  0.90879 

hV={-)    log  1.58066 

A7  =  100    locr  2.00000 colog  8.00000-10 


Ai?M,'=(-)294.53   log  2.46914 

5  =  .123372 

4  =  .066658 


£-.4  =  .056714   log  8.75369-10 

i^  =  19000    log  4.27875 

5  =  . 12337 colog  0.90879 

A«=+10 log  1.00000 

m;  =  260   colog  7.58503-10 

Ai?,„"=(  +  )335.94    log  2.52626 

Ai?,,=  (  +  )   41.44 

hence  an  increase  in  weight  gives  an  increase  in  range  for  this  gun  at  this  range,  and 
this  quantity  would  carry  a  negative  sign  in  Column  11  of  the  range  table  for  this 
range. 

Note  to  Form  No.  13. — The  method  of  Case  2  is  of  course  the  more  accurate,  and  gives 
practically  the  range  table  results.  The  method  shown  in  Case  1  is  introduced  to  give 
practice  in  the  use  of  these  formulae  without  the  necessity  for  taking  up  the  successive 
approximation  method  in  order  to  determine  the  value  of  G  accurately. 


294  APPENDICES 


Form  No.  14. 

CHAPTEE  12— EXAMPLE  5. 

FORM  FOR  THE  COMPUTATION  OF  THE  CHANGE  IN  RANGE  RESULTING 
FROM  A  VARIATION  FROM  STANDARD  IN  THE  WEIGHT  OF  THE  PRO- 
JECTILE, OTHER  CONDITIONS  BEING  STANDARD.  SHORT  METHOD, 
USING  DATA  CONTAINED  IN  COLUMNS  10  AND  12  OF  THE  RANGE 
TABLES. 

FOEMUL^. 

87  =  0.36  ^^  Y;  ^E^  =  ^EJ  +  ^EJ'  =  ^RYX  S  +^  XAScXAS 
w  oV  w 

PEOBLEM. 

Cal.  =  8";  7  =  2750  f.  s.;  w  =  260  pounds;  c  =  0.61;  Eange  =  19,000  yards;  Erom 
Column  10  of  range  table,  Ai25oy  =  387  yards;  From  Column  12  of  range  table, 
AEioO=874  yards;  Variation  in  weight=  +10  pounds. 

Aw=  +10    log  1.00000 

w;  =  260   colog  7.58503-10 

7  =  2750   log  3.43933 

.36 log  9.55630-10 


hV={-)    log  1.58066 

AJ?5of  =  387    log  2.58771 

87=(-)    (-)log  1.58066 

87'  =  50    colog  8.30103-10 

A7?^'=  (-)294.71   (  -  )log  2.46940 

Ai?^oC=  874    log  2.94151 

Aw=  +10   log  1.00000 

w  =  260   colog  7.58503-10 

A8=  10   lo^  1.00000 


Ai2«;"=(  +  )336.15    log  2.52654 

Ai^i<,=  (  +  )    41.44  yards, 
hence  an  increase  in  weight  gives  an  increase  in  range  for  this  gun  at  this  range, 
therefore  this  quantity  would  carry  a  negative  sign  in  Column  11  of  the  range  table 
at  this  range. 

Note  to  Form  No.  14. — The  method  gives  practically  the  range  table  result. 


APPENDICES  295 


Form  No.  15. 

CHAPTER  12— EXAMPLE  6. 

FORM  POR  THE  COMPUTATION  OF  THE  CHANGE  IN  THE  VERTICAL  POSI- 
TION OF  THE  POINT  OF  IMPACT  IN  THE  VERTICAL  PLANE  THROUGH 
THE  TARGET  RESULTING  FROM  A  VARIATION  IN  THE  SETTING  OF  THE 
SIGHT  IN  RANGE,  ALL  OTHER  CONDITIONS  BEING  STANDARD. 

,     FORMULA. 
H  =  AX  tan  0) 

PROBLEM. 

Cal.  =  S";  y  =  2750  f.  s.;  iv  =  260  pounds;  c  =  0.61;  Range  =  19,000  yards; 

u)  =  26°  35'  00"  (from  range  table) ;  Variation  in  setting  of  sight=  +150  yards. 

AZ  =  450   log  2.65321 

<o  =  26°   35'  00"    tan  9.69932-10 

H=  +225.18  feet   log  2.35253 


296  APPENDICES 


Form  No.  16. 

CHAPTEE  13— EXAMPLE  1. 
FORM  FOR  COMPUTATION  OF  THE  DRIFT. 

FORMULA. 
D=-t^x~  X  -^^  X  Multiplier 

PEOBLEM. 

Case  1. — Correcting  for  Altitude  by  Table  V. 

Cal.  =  8";  7  =  2750  f.  s.;  w  =  260  pounds;  c=0.61;  Eange=19,000  yards  =  57,000 
feet;  <^  =  15°  07'  00"  (from  range  table)  ;  Maximum  ordinate  =  5261  feet  (from 
range  table)  ;  fF  =  3507  feet. 

K=   log  0.82346 

/=1.0962    log  0.03989 

C=    log  0.86335 colog  9.13665-10 

Z  =  57000   log  4.75587 


Z  =  7807.6   log  3.89252 

i)'  =  484+  ^^^  _  3^^  =466.67 

Z>'  =  466.67   log  2.66901 

Multiplier  =  1.5   log  0.17609 

^=.53  log  9.72428-10 

n  =  25   log  1.39794 colog  8.60206-10 

A  =.32    log  9.50515-10 

h 

C=    log  0.86335 2  log  1.72670 

<^  =  15°  07'  00"    sec  0.01529 3  sec  0.04587 


2>  =  281.29  yards log  2.44916 

PEOBLEM. 

Case  2. — Using  Corrected  Value  of  C  and  (f>  Obtained  by  Successive  Approximations 

on  Form  No.  1. 
Cal.  =  8";  7  =  2750  f.  s.;  w  =  260  pounds;  c=0.61;  Eange  =  19,000  yards  =  57,000 
feet;  log  (7  =  0.87827;  «^  =  15°  07'  15"  (from  Form  No.  1). 

C=    colog  9.12173-10 

Z  =  57000   log  4.75587 


Z  =  7544.0   log  3.87760 

r>,_4oo 4.  20>^  _  33X50  _ 
^-^^"^+"^00"     "100^-^^^-'^ 


APPENDICES  297 

D'  =  4U.3 log  2.61731 

Multipliers  1.5   log  0.17609 

fi  =  .53  log  9.72428-10 

n  =  25  log  1.39794 colog  8.60206-10 

^=.32    log  9.50515-10 

h  ° 

C=    log  0.87827 2  log  1.75654 

<^=15°  07'  15"    sec  0.01530 3  sec  0.04590 


Z)  =  267.50  yards log  2.42733 

Note  to  Form  No.  16. — The  method  of  Case  2  is  of  course  the  more  accurate,  and  gives 
practically  the  range  table  result.  The  method  shown  in  Case  1  is  introduced  to  give 
practice  in  the  use  of  this  formula  without  the  necessity  for  taking  up  the  successive 
approximation  method  in  order  to  determine  the  exact  values  of  C  and  </>. 


Form  No.  17. 

CHAPTER  13— EXAMPLE  2. 

FORM  FOR  COMPUTATION  OF  SIGHT  BAR  HEIGHTS  AND  SETTING  OF 

SLIDING  LEAF. 

(Permanent  Angle  =  0°.) 

FOEMUL.^. 

h  =  ltan<t>  d=^-^^D 

K 

PROBLEM. 

Cal.  =  8";  7  =  2750  f.  s.;  w  =  260  pounds;  c  =  0.61;  Range  =  19,000  yards; 
</)=15°  07'  00";  Sight  radius  =  41.125";  Deflection  =  266  yards  right. 

<^  =  15°   07'  00" tan  9.43158-10..     sec  0.01529 

Z  =  41.125    log  1.61411 log  1.61411 

i2  =  19000    log  4.27875 colog  5.72125-10 

Z)  =  266 W  2.42488 


/i  =  11.110"   W  1.04569 


(f  =0.59639"  left log  9.77553-10 


298  APPENDICES 

Form  No.  18. 

CHAPTEE  14— EXAMPLE  1. 

FORM  FOR  THE  COMPUTATION  OF  THE  EFFECT  OF  WIND. 

FOEMUL.^. 

^  _  17x6080         _  F"  sin  2<^ 
^^"- 60X00X3'''-        gX        ' 

^E^=^yJT-  -JL.  x  ^^);  d^=wJt-  ^^) 

\        2n  —  l  V     /  \         7cos^/ 

PEOBLEM. 

Cal.  =  S":  r  =  2750  f.  s. ;  w  =  260  pounds;  c  =  0.61;  Eange  =  19,000  yards  =  57,000 
feet;  <^  =  15°  07'  00";  T=35.6  seconds  (</>  and  T  from  range  table)  ;  Wind  com- 
ponent along  line  of  fire  =  15  knots  an  hour  with  the  flight;  Wind  component 
perpendicular  to  the  line  of  fire  =  10  knots  an  hour  to  the  right. 

15    log  1.17609 

10    ' log  1.00000 

6080    log  3.78390 log  3.78390 

60x60x3  =  10800    colog  5.96658-10.  .colog  5.96658-10 

Wr=    log  0.92657 


Wz=    log  0.75048 

7  =  2750    2  log  6.87866 

2(^  =  30°   14'  00"   sin  9.70202-10 

Z  =  57000 log  4.75587 colog  5.24413-10 

^  =  32.2  colog  8.49214-10 

n  =  2.0746   log  0.31695 

2w  =  4.1492 

2n-l  =  3.1492    log  0.49820 colog  9.50180-10 

71  =  2.0746   log  0.31695 

</)  =  15°  07'  00    cos  9.98471-10 

Z  =  57000   log  4.75587 

7  =  2750   coloff  6.56067-10 


13.182    log  1.12000 

T=35.640 


22.458    log  1.35137 

W^=    log  0.92657 


Ai?TF=  189.64  yards  over log  2.27794 

Z  =  57000    log  4.75587 

7  =  2750 colog  6.56067-10 

«/.  =  15°   07'  00"   sec  0.01529 

21.47    log  1.33183 

r=35.64 


14.17 log  1.15137 

W,=    log  0.75048 


Dw  =  79.771  yards  right log  1.90185 


APPENDICES  299 

Form  No.  19. 

CHAPTER  14— EXAMPLE  2. 

FORM  FOR  THE  COMPUTATION  OF  THE  EFFECT  OF  THE  MOTION  OF  THE  GUN. 

FORMULAE. 


«  _    Gx6080    .     ^  V^  sin  2<l> 
^      60  X  60  X  3  '  "  gX       ' 

n      ^  Z  cos  <^  ^  .  ^   _ 

V  cos  <^ 


60  X  60  X  3  '  gX 

ap  —      ^      V,  -ST cos <^  r  •  n  —       ^      n 


PROBLEM. 

Cal.  =  8";  7  =  2750  f.  s.;  w;  =  260  pounds;  c  =  0.61;  Range  =  19,000  yards  =  57,000 
feet;  ^  =  15°  07'  00"  (from  range  table)  ;  Speed  component  in  line  of  fire  =  9 
knots  an  hour  against  the  flight;  Speed  component  perpendicular  to  the  line  of 
fire  =  18  knots  an  hour  to  the  left. 

9   log  0.95424 

18    log  1.25527 

6080    log  3.78390 log  3.78390 

60x60x3  =  10800    colog  5.96658-10.  .colog  5.96658-10 

G:.=   log  0.70472 


Gz-    log  1.00575 

7  =  2750   2  log  6.87866 

2(^  =  30°   14'  00"  sin  9.70202-10 

g  =  32.2   colog  8.49214-10 

Z  =  57000   loff  4.75587 colog  5.24413-10 


n  =  2.0746 log  0.31695 

2n  =  4.1492 

2n-l  =  3.1492    log  0.49820 colog  9.50180-10 

n  =  2.0746   log  0.31695 

Z  =  57000   log  4.75587 

</>  =  15°  07'  00"  cos  9.98471  - 10 

7  =  2750   colog  6.56067-10 

G^=    log  0.70472 

A/?G  =  66.791  yards  short log  1.82472 

Z  =  57000   log  4.75587 

7  =  2750   colog  6.56067-10 

<^  =  15°   07'  00"  sec  0.01529 

Gz=    log  1.00575 

Z)o  =  217.56  yards  left log  2.33758 


300  APPENDICES 


Form  No.  20. 

CHAPTEE  14— EXAMPLE  3. 

FORM  FOR  THE  COMPUTATION  OF  THE  EFFECT  OF  MOTION  OF  THE  TARGET. 

FORMULA. 

^^=Sf-3'^^^-^^^^-^^  =  ^^^ 

PROBLEM. 

Cal.  =  8";  y  =  2750  f.  s.;  w  =  2Q0  pounds;  c=0.61;  Range  =  19,000  yards:  Time  of 
flight^ 35.64  seconds  (from  range  table)  ;  ^a;  =  Speed  component  in  line  of  fire 
in  knots  per  hour  =17  knots  with  flight;  >S'«  =  Speed  component  perpendicular  to 
line  of  fire  in  knots  per  hour  =  19  knots  to  left. 

r  =  35.64   log  1.55194 log  1.55194 

S:,  =  17    log  1.23045 

Sz  =  19   log  1.27875 

6080 log  3.78390 log  3.78390 

60x60x3  =  10800    colog  5.96658-10.  .colog  5.96658-10 

Ai2r=  341.09  yards  over log  2.53287 

Dt  =  381.22  yards  right log  2.58117 

Note  to  Form  No.  20. — Note  that  this  example  is  simply  the  arithmetical  problem  of 
determining  how  far  the  target  will  move  in  the  given  direction  at  the  given  speed  during 
the  time  of  flight;  the  speeds  being  given  in  knots  per  hour,  and  the  results  required  in 
yards  for  the  time  of  flight. 


APPENDICES  301 


Form  No.  21. 

CHAPTER  16— EXAMPLE  1. 

FORM  FOR  THE  COMPUTATION  OF  THE  PENETRATION  OF  HARVEYIZED  (E^] 
AND  OF  FACE-HARDENED  {£,)   ARMOR  BY  CAPPED  PROJECTILES. 

EORMUL.^. 
Harveyized  Armor  (Davis).  Face-Hardened  Armor  (De  Marre). 

p  0.8 _  vw^-^  -p  o.7_  vw"-"-  "" 

"^1         ~    E-'JO.S  -^2         —    T' J0.75    '^ 


K'd''-^  -         A'cZ"'-^^       De  Marie's  Coefficient 

log  A"  =  3.25313  log  A  =  3.00945 

PEOBLEM. 

Cal.  =  8";  F  =  2750  f.  s.;  w  =  260  pounds;  c  =  0.61;  Eange  =  19,000  yards;  r„  =  1184 
f.  s.  (from  range  table) ;  De  Marre's  coefficient=1.5. 

w  =  260    log  2.41497 0.5  log  1.20748 

i;^  =  llS4    log  3.07335 


v„w°-^=    log  4.28083 log  4.28083 

K'=    colog  6.74688-10 

K=    colog  8.99055-10 

d  =  S    W  0.90309..  0.5  colog  9.54846-10.  .0.75  colog  9.32268-10 


i;i°-«=    log  0.57617 

10 

8  I  log  5.76170 
^1  =  5.2506"  log  0.72021 


log  0.59406 
10 


y  I  log  5.94060 
log  0.84866 
De  Marre's  coefficient  =  1.5 coloff  9.82391-10 


.£^2  =  4.7051" log  0.67257 


Form  No.  22.  CHAPTER  16— EXAMPLE  2. 

FORM  FOR  THE  COMPUTATION  OF  <^,  «,  f,  /„,  D,  Y,  AND  THE  PENETRATION, 
GIVEN  R  AND  /;  ATMOSPHERIC  CONDITIONS  STANDARD.* 
FOEMUL.^. 
C=-^-^  =fK;  ^  =  ~p  J  sin  2c;6  =  x4C;  tan  m  =  B'  tan  0;  t)£j  =  w£j  cos  </>  sec  w 


w/i        cos''  </> 


C 


-^1^'*—  7^/^0.5  (Harveyized — Davis) 


r  =  A"C  tan  0;  £'2°-'=  /^-^o.^r    ;^  5  (Face  hardened— De  Marre) 


Cal.  =  8";  7  =  2750  f.  s.;  tv  =  2Q0 
feet;  /=  1.1345. 

log  A'    0.82346 

log/    0.05481 

logC    0.87827 

cologC 9.12173- 

logJC 4.75587 


PROBLEM. 

pounds;  c  =  0.61;  Range  =  19,000  yards  =  57,000 


10 


logZ    3.87760 

Z  =  7544.0 

(2)    9^  =  15°  07' 15" 

logC    0.87827 

log  A    8.82386 


log  sin  20    9.70213 

2<f>  =  30°  14'  30" 

(3)   cu  =  26°  34' 40" 

logB'    0.26751 

log  tan  0    9.43170 


log  tan  cu    9.69921 

(4)  T  =  35.642  seconds 

logC    0.87827 

logT'    0.65839 

log  sec  0    0.01530 

log  T    1.55196 

(5)  t;„  =  1184.2 

logM^,     3.04021 

log  cos  0 9.98470 

log  sec  o)    ' 0.04850 


^0 
10 


To 


10 


logv,,,     3.07341 

(6)   23  =  267.50 

log  At  (.53)    9.7242S 

cologn(25)     8.60206 

log  ^-(.32)     9.50515 


log  constant  7.83149 

logC^  1.75654 

log  sec'  0  0.04590 

logD'  2.61731 

2.25124 
log  1.5  (if  used) 0.17609 

logD  2.42733 


10 
10 

10 

"10 


A  =  .06851  + 


.00170  X  44 


100 
00520  X  50 


100 
=  2465-^X 


=  .066658 
(—.0048)  X  68 


50  X  0 
"^  100  "^ 
=  2578.1 


.0025 
001758  X  68 
0025 


log  B'  =  . 2652  + 


.0023  X  44 


100 
.0026  X  50 


T'  =  4.600  + 


100 

.092  X  44 


=  .267512 

173  X  50 


100 


100 


=  4.55398 


u,„  =  1086  — 


9  X  44  ,  30  X  50 


D'  =  422  + 


100 
=  1097.04 

20  X  44 


+ 


100 


33  X  50 


100 


100 
=  414.30 

(8)  Y  =  5263.4  feet 

log  A"  3.41130 

logC  0.87827 

log  tan  0  9.43170  —  10 

logY  3.72127 

(9)  Harveyized  armor.     £1  =  5.2515  in. 

logio"-'*     1.20748 

colog^'   6.74688  —  10 

colog  d""'    9.54846  —  10 

logi?^     3.07341 

logE°-^    0.57623 

logE,    0.72028 

(9)   Face  hardened,     i:^  =  4.706  in. 

logw"-'*     1.20748 

colog  ^    6.99055 

colog  fZ"-"   9.32268  —  10 

logVy      3.07341 

log£?2»'    0.59412 

log(B,  X1.5)     0.84874 

colog  1.5    9.82391 


10 


10 


logE,    0.67265 

RESULTS. 

0  =  15°  07'  15"  D  =  267.50  yards. 

w  =  26°  34'  40".  Y  =  5263.4  feet. 

T  =  35.642  seconds.      E^  =  5.252  inches. 

r„,  =  1184.2  f.  s.  E„  =  4.706  inches. 


*  If  we  have  a  problem  in  vv^hich  /  is  not  known,  then  we  must  first  determine  the 
value  of  0  for  the  given  range  by  the  use  of  Form  No.  1  in  paragraph  273,  Chapter  16. 


APPENDICES 


303 


Form  No.  23. 

CHAPTER  16— EXAMPLE  3. 

FORM  FOR  THE  COMPUTATION  OF  S,  ARy,  ^Rc,  A/?^,  FOR  A  GIVEN  R  AND  f. 


FORMULA. 


^=  i^ =/^'-  ^=  # 


(g-^)B 


\ 

^;   87  =  0.36^7; 
8  w 


l+-^cot 


>^^y=-B^^Yv^^' 


AR^  =  ARu,'  +  ^Rw"  =  ^RvX   f^^  +  —  XA7?5XA8 

o  K  it' 


Cal.  =  8";  7  =  2750  1  s.;  w  =  260 
feet;</)  =  15°  07'  15";o>  =  26° 

log£:    0.82346 

log/   0^05*^ 

logC    0787827 

cologC    9.12173 

logX    4.75587 


PROBLEM. 

pounds;  c  =  0.61;  Range  =  19,000  yards  =  57,000 
3-i'  40";  /=  1.1345 ;  /i  =  20  feet;  Aw=±5  pounds. 

.00012  X44 


10 


logZ     3.87760 

Z  =  7544.0 

(7)   /S  =  13.335  yards 

logy  (6.6667)    0.82391 

log  cot  a.    0.30079 

log  (6.6667  cot  a>)     1.12470 

cologT?    5.72125 

log  ^'^^^^  cot  u.    6.84595 


—  10 


R 


—  10 


6.6667 
R 


cot  03  =  .00070 


,      6.6667       , 
1  + — J.—    cot  w 

=  1.0007    log  0.00030 

6.6667  cot  w    log  1.12470 

logs  1.12500 

(10)   A/?,  =386.77  yards 

logAr^     7.70094 

cologS    0.90879 

log5F(50)     1.69897 

cologAy(lOO)    8.00000- 

logR    4.27875 


10 


10 


log  ^R,,r 2.58745 

RESULTS. 
/S  =  13.335  yards. 
^R^y=  386.77  yards. 
Ai?,o5  =  873.43  yards. 
Ai?jo  =  ±  20.70  yards. 


Af^  =  .00520  + 


100 
00046  X  50 


A  =  .06851  + 


100 

.00170  X  44 


=  .0050228 


100 
00520  X  50 


B  =  .1261  + 


100 
.0038  X  44 


=  .066658 

0088  X  50 


100  100 

=  .123372 

(12)   A7?5  =873.43  yards 

log  (B  —  A )    8.75369  —  10 

log  7?    4.27875 

colog  B    0.90879 

log  (0.1)     9.00000  —  10 

logA/?5 2.94123 

(11)   A«,„  =  ±  20.70  yards 

log  Aw    0.69897 

colog  w    7.58503  —  10 

log  V   3.43933 

log  0.36    9.55630  —  10 

logSF  1.27963 


logA/?,„v  2.58745 

logSr 1.27963 

colog  5F'(50)    8.30103  —  10 

log  AT?,/    2.16811 


logA7?i„5 2.94123 

log  Aiy    0.69897 

colog  10    7.58503  —  10 

logA5(10)     1.00000 

log  ARu" 2'!22523 

AR^'  =  If:  147.27 
ARy,"  =  ±  167.97 

A7?,t,  =  ±:    20.70 

An  Increase  in  weight  gives  an  increase 
in  range  for  this  gun  at  this  range,  there- 
fore this  quantity  would  carry  a  negative 
sign  in  Column  11  of  the  range  table  for 
this  range. 


504 


APPEXDICES 


Form  No.  24. 

CHAPTER  16— EXAMPLE  4. 

FORM  FOR  THE  COMPUTATION  OF  EFFECTS  OF  WIND  AND  OF  MOTION  OF 
GUN  AND  TARGET;  ALSO  CHANGE  IN  HEIGHT  OF  POINT  OF  IMPACT 
FOR  VARIATION  IN  SETTING  OF  SIGHT  IN  RANGE  FOR  A  GIVEN  R  AND  /. 


FORMULA. 


gX  \         2n  —  l  VI  \         7cos^/' 

\2n  —  l  V      /  Vcostfi 


Dt=T,T;  ]Fx  =  Etc.= 


FX6080 
3x60x60 


PROBLEM. 

Cal.  =  8";  7  =  2750  f.  s.;  ?(7  =  260  pounds;  c  =  0.61;  Range  =  19,000  yards  =  57,000 
feet;  <^  =  15°  07'  15";  r  =  35.642  seconds;  (o  =  26°  34'  40". 


Value  of  n 

logV^    6.87866 

log  sin  20   9.70213  —  10 

cologgf(32.2)    8.49214  —  10 

cologZ    5.24413  —  10 

logn    0.31706 


2n 


n  =  2.0752 
2n  =  4.1504 
-1  =  3.1504 


(13)   ARw  =151.74  yards 

logn    0.31706 

logZ    4.75587 

log  cos  0    9.98470  —  10 

colog(2n  — 1)    9.50163  —  10 

colog  V    6.56067  —  10 

■"s^l^^w "^^^ 

T  =  35.642 

log    22.462 1.35145 

logWa; 0.82966 

logARw 2.18111 

(16)   Dw  =95.740  yards 

logZ    4.75587 

colog  y    6.56067  —  10 

log  sec  0    0.01530 

X 


.1.33184 


V  cos  <p 
T 


21.470 
35.642 


log    14.172 1.15143 

log  W^    0.82966 

\ogDw     1.98109 


Value  of  Wj,,  W^.  Gx,  G^,  T^,  T«  for  a 
component  of  12  knots. 

log  12    1.07918 

log  6080   3.78390 

colog  3    9.52288  —  10 

colog(60)^    6.44370  —  10 

logWj.,  etc 0.82966 

(14)   ARa  =89.040  yards 
nX  cos  9 


log 


,1.11993 


"°  (2n  — l)y    

log  Gj    0.82966 

log  ARa    1.94959 

(17)   Do  =  145.04  yards 


X 


.1.33184 


^°°  V  cos  4> 

logG^     0.82966 

log  Da    2.16150 

(15)   Ai2T  =  D2  =  240.78  yards 

log  Ta,  =  T«    0.82966 

logT    1.55196 

log  AR  T  =D  T 2.38162 

(19)  H=  150.08  feet 

log  Z  =  300    2.47712 

log  tan  to    9.69921  —  10 

log  if  100    2.17633 

RESULTS. 
n  =  2.0752. 
ARw  =  151.74  yards. 
Dw  =    95.740  yards. 
ARa  =    89.040  yards. 
Do  =  145.04  yards. 
ARt  =  Dt  =  240.78  yards. 
fl'  =  150.08  feet. 


APPENDICES 


305 


Form  No.  25A. 

CHAPTER  17— EXAMPLE  8. 

FORM  FOR  THE  SOLUTION  OF  REAL  WIND  AND  SPEED  PROBLEMS. 


PEOBLEM. 

Cal.  =  8";  F  =  -2750  f.  s.;  iv  =  2C,0  pounds;  c  =  O.Gl;  Eange  =  7500  yards;  Peal  wind, 
direction  from,  225°  true;  velocity,  15  knots  an  hour;  Motion  of  gun,  course, 
355°  true;  speed,  20  knots  an  hour;  Motion  of  target,  course,  330°  true;  speed, 
25  knots  an  hour;  Target  at  moment  of  firing,  75°  true;  distant,  7500  yards; 
Barometer  =  30.00";  Thermometer  =  10°  F.;  Temperature  of  powder  =75°  F.; 
Weight  of  projectile  =  263  pounds. 


—  35 
Temperature  of  powder,  — j-^   X  15 : 


52.5  foot-seconds. 


Use  Table  IV  to  correct  for  density.    Use  traverse  tables  for  resolution  of  wind 
and  speeds. 


1/77^  o/"  F/r^ 


Wind 


20 


306 


APPENDICES 


Cause  of  error. 
Speed  of  or  varia-       Affects. 
tion  in. 


Gun 


Tarsret 


Wind 


Initial  velocity  . . 


Density. 


Range .... 
Deflection. 
Range .... 
Deflection. 
Range .... 
Deflection. 
Range .... 
Range .... 


Range . 


12 


Formulae. 


44         3.5  X  44 


12  ~ 

12 

20  sin  80  X 

55 

12  ~ 

in.7  X  55 
12 

25  cos  65  X 

68 

10.6  X  68 

12  ~ 

12 

25  sin  65  X 

68 

22.7  X  68 

12 

12 

15  cos  30  X 

24 

13  X  24 

12 

12 

15  sin  30  X 

13 

7.5  X  13 

12 

12 

207 
52.5  X^ 


3X 


43 


1.25  X  180 


30.2  yards  X  ~?ro=5-3  knots  on  deflection  scale. 
68 


Errors  in — 


Range. 


Short. 
Yds. 


217.4 

25.8 
225.0 


468.2 
98.9 


369.3 


Over. 

Yds. 


12.8 


60 


26 


98.9 


Deflection. 


Right. 
Yds. 


128.6 


128.6 
98.4 


30.2 


Left. 
Yds. 


90 


Set  sights  at: 

Exactly in  range,  7869.3  yards;   in  deflection,  44.7  knots. 

Actually in  range,  7850.0  yards;  in  deflection,  45.0  knots. 

Remember  to  shoot  short  rather  than  over. 


APPENDICES 


307 


Form  No.  25B. 

CHAPTER  17— EX A:\rPLE  8. 

FORM  FOR  THE  SOLUTION  OF  REAL  WIND  AND  SPEED  PROBLEMS. 


PROBLEM. 

Cal.  =  8";  7  =  2750  f.  s.;  w  =  260  pounds;  c  =  0.61;  Range  =  7000  yards;  Real  wind, 
direction  from,  160°  true;  velocity,  20  knots  an  hour;  Motion  of  gun,  course, 
2()0°  true;  speed,  18  knots  an  hour;  Motion  of  target,  course,  170°  true;  speed, 
23  knots  an  hour;  Target  at  moment  of  firing  bearing,  115°  true;  distant,  7000 
yards;  Barometer  =  29.00";  Thermometer  =  85°  E.;  Temperature  of  powder 
=  95°  F. ;  Weight  of  projectile  =  258  pounds. 


Temperature  of  powder. 


+  35 
10 


X  5  =  + 17.5  foot-seconds. 


Use  Table  IV  for  correction  for  density, 
speeds. 


Use  traverse  tables  for  resolving 


^n 


308 


APPENDICES 


Cause  of  error. 
Speed  of  or  vari- 
ation in. 


Gun. 


Target. 


Wind, 


Initial  velocity 


Density. 


Affects. 


Range 

Deflection.. 

Range 

Deflection.. 

Range 

Deflection.. 
Range 


Range , 
Range . 

12 


FormulsB. 


4->        14  7  V  4'> 
18  cos  35  x^  = 


12 


12 


-^     .      „.  ^  52        10.3X52 
18  sin  do  X  -vv  = 


12 


12 


63        13.2  X  03 
23  cos  55  X  ^  = ^ — 


23  sin  55  X 

63 

_  18.8  X  63 

12 

12 

20  cos  45  X 

21 
U 

14.1  X  21 
12 

20  sin  45  X 

11 

14.1  X  11 

12 

12 

17.5  X 


197 
"50 


43 
.69  X  157 


67  yards  X  "^=  12.  8  knots  on  deflection  scale. 


Errors  in  — 


Range. 


Short. 
Yds. 


51.5 


69.3 


24.7 


145.5 


Over. 
Yds. 


69.0 

17.2 

108.3 


194.5 
145.5 


49.0 


Deflection. 


Right. 
Yds. 


Left. 

Yds. 


44 


44. G 


98.7 


12.9 


111.6 
44.6 

67.0 


Set  sights  at : 

Exactly in  range,  6951  yards ;  in  deflection,  62.8  knots. 

Actually in  range,  6950  yards;  in  deflection,  62.0  knots. 

Remember  to  shoot  short  rather  than  over. 


APPENDICES 


309 


Form  No.  26. 

CHAPTER  17— EXA]\rPLE  9. 

FORM  FOR  THE  SOLUTION  OF  APPARENT  WIND  AND  SPEED  PROBLEMS. 


PEOBLEM. 

Cal.  =  8";  7=2750  f.  s.;  w  =  2G0  pounds;  c  =  0.61;  Range  =  7300  yards;  Apparent 
wind,  direction  from,  45°  true;  velocity,  30  knots  an  hour;  Motion  of  gun, 
course,  80°  true;  speed,  21  knots  an  hour;  Motion  of  target,  course,  100°  true; 
speed,  28  knots  an  hour;  Target  at  moment  of  firing  bearing,  300°  true;  distant, 
7300  yards;  Barometer  =  28.50";  Thermometer  =  10°  F.;  Temperature  of  powder 
=  60°  F. ;  Weight  of  projectile  =  255  pounds. 

Temperature  of  powder,         -  x  30=  — 105  foot-seconds. 

Use  Table  IV  for  correction  for  density.    Use  traverse  tables  for  resolution  of 
speeds. 


Ji^'ziC^ 


l/'ve  of 
f}'re 


310 


APPENDICES 


Cause  of  error. 
Speed  of  or  vari- 
ation in. 


Gun. 


Target . 


Wind 


Initial  velocity. 


Density. 


Affects. 


Range 

Deflection.. 

Range 

Deflection. . 

Range 

Deflection.. 

Range 

Range 


Formulae. 


21  cos  40  X 


66        16.1  X  66 


12 


12 


66        13.5  X  66 
21  sin  40  X  -j^  ^ 


12 


66        26.3  X  66 
28  cos  20  X  j^~  = ^2 

66  9.6  X  66 


ilO    sill    -V   /\ 

12 

12 

30  cos  75  X 

23 

7.8  X  23 

12 

12 

30  sin  75  X 

12  _ 

29  X  12 

12 

12 

Range . 


105  X 


203 
50" 


43  X -5 
.69  X  171 


12 


7.5  yards  deflection  X  -7t^=  1-4  knots  on  deflection  scale. 


66 


Errors  in — 


Range. 


Short. 
Yds. 


88.6 


426.3 


118.0 


632.9 
202.6 


430.3 


Over. 
Yds. 


144.7 


14.9 


Deflection. 


Right. 
Yds. 


74 


43.0 


202.6 


74 


Set  sights  at: 

Exactly in  range,  7730.3  yards;   in  deflection,  51.4  knots. 

Actually in  range,  7700.0  yards;   in  deflection,  51.0  knots. 

Remember  to  shoot  short  rather  than  over. 


Left. 
Yds. 


52.8 


29.0 


81.8 
74.3 

7.5 


APPENDICES 


311 


Form  No.  27. 

CHAPTER  18— EXAMPLES  1  AXD  2. 

FORM  FOR  THE  COMPUTATIONS  FOR  THE  CALIBRATION  OF  A  SINGLE  GUN 
AND  FOR  DETERMINING  THE  MEAN  DISPERSION. 


PROBLEM. 

Cal.  =  8";  7  =  2750  f.  s.;w  =  260  pounds;  c= 0.61 ;  Actual  range  =  8500  yards;  Sights 
set  at,  in  range,  8500  yards;  in  deflection,  40  knots;  Center  of  bull's  eye  above 
water,  12  feet;  Bearing  of  target  from  gun,  37°  true;  Wind  blowing  from  350° 
true,  with  velocity  of  8  knots  an  hour;  Barometer= 29.85";  Thermometer 
=  80°  P.;  Temperature  of  powder=100°  P.;  Weight  of  shell  =  268  pounds; 
iSTumber  of  shot  =  4,  falling  as  follows: 


Range. 

Deflection. 

Shot. 

Short. 
Yds. 

Over. 
Yds. 

Right. 
Yds. 

Left. 
Yds. 

1 

100 
125 

85 
90 

85 

9 

80 

3 

60 

4 

55 

Mean   errors  on  foot 
of  perpendicular 
through  bull's  eye . .  . 

41400.0 
100.0  short 

4|280 

70.0  left 

Correction  in  range  due  to  height  of  bull's  eye : 

^X  12  =  38.7  yards  over 
31  •' 

Correction  in  deflection  due  to  intentional  displacement: 
-||x  10  =  66.7  yards  left 

Temperature  of  powder: 

±M  X 10  =  +  35  foot-seconds 

Use  Table  IV  for  density  correction  and  traverse  tables  for  resolution  of  wind 
forces. 


313 


APPENDICES 


^md 


Lme  of 
F/'re 


TA/ZQ£rT 


Short. 
Yds. 

Over. 
Yds. 

Right. 
Yds. 

Left. 

Yds. 

Wind - 

Range 

Deflection. . 

Range 

Range 

Range 

Range 

Deflection.. 

31        5.5  X  31 
8cos47Xi2-       12 

8sin47x;^--f^ 

«x^ 

.318  X  229 

227 
='X50 

-x^» 
'°xS 

14.2 
65.6 

.... 

72.8 
158.9 

38.7 

8.4 

.... 

Density 

Initial  velocity  . . . 

Height    of    bull's 
eve 

.... 

Intentional    deflec- 
tion  

66.7 

79.8 

270.4 
79.8 

190.6 

8.4 

66.7 
8.4 

Errors  on  point  P  as  an  origin  i 

or  standard  conditions 

58.3 

Observed  distance  from  target  in  range 100.0  yards 

Error   (where  shot  should  have  fallen) 190.6  yards 


short, 
over. 


True  mean  error  in  range  under  standard  conditions 290.6  yards  short. 

Observed  distance  from  target  in  deflection 70.7  yards  left. 

Error  (where  shot  should  have  fallen) 58.3  yards  left. 


True  mean  error  in  deflection  under  standard  conditions.  .    11.7  yards  left. 


APPENDICES 


313 


That  is,  under  standard  conditions  the  mean  point  of  impact  of  the  gun  is  290.6 
yards  short  of  and  11.7  yards  to  the  left  of  the  point  P.  We  wish  to  adjust  the  sight 
scales  so  that  the  actual  mean  point  of  impact  of  the  gun  shall  be  at  P.    To  do  this  we : 

1.  Eun  up  the  sight  until  the  pointer  indicates  8790.6  yards  in  range,  then  slide 
the  scale  under  the  pointer  until  the  latter  is  over  the  850"0-yard  mark  on  the  former, 
and  then  clamp  the  scale. 

12 

2.  11.7  yards  in  deflection  equals -x^  X  11.7  =  1.8  knots  on  the  deflection  scale 

at  the  given  range.    Set  the  sight  for  a  deflection  of  51.8  knots,  then  slide  the  scale 
until  the  50-knot  mark  is  under  the  pointer,  and  then  clamp  the  scale. 


MEAN  DISPERSION  FROM  MEAN  POINT  OF  IMPACT. 


In  range. 

In  deflection. 

Number  of  shot. 

Fall  relative 

to  target. 

Short  or 

over. 

Yds. 

Position  of 

mean  point 

of  impact 

relative  to 

target. 

Short  or 

over. 

Yds. 

Variation  of 
each  shot 
from  mean 
point  of  im- 
pact.    Short 
or  over. 
Yds. 

Fall  relative 

to  target. 

Right  or 

left. 

Yds. 

Position  of 

mean  point 

of  impact 

relative  to 

target. 

Right  or  left. 

Yds. 

Variation  of 

each  shot 

from  mean 

point  of 

impact. 

Right  or  left. 

Yds. 

1 

100  short 

125  short 

8.5  short 

90  short 

100  short 
100  short 
100  short 
100  short 

0 
25 
15 
10 

85  left 
80  left 
60  left 
55  left 

70  left 
70  left 
70  left 
70  left 

15 

2 

10 

3 

10 

4 

15 

4|50 
12.5 

4|50 
12.5 

Mean  dispersion  from  mean  point  of  impact: 

In  range 12.5  yards. 

In  deflection 12 . 5  yards. 


314 


APPENDICES 


Form  No.  28. 

CHAPTER  19— EXAMPLE  1. 

FORM  FOR  THE  COMPUTATIONS  FOR  THE  CALIBRATION  OF  A  SHIP'S 

BATTERY. 


PEOBLEM. 
Cal.  =  8";  7  =  2750;  w  =  260  pounds;  c  =  0.61;  Eange  =  8500  yards. 

For  a  battery  of  eight  of  the  above  guns,  having  determined  the  true  mean  errors 
to  be  as  given  below  (by  previous  calibration  practice),  how  should  the  sights  of  the 
guns  be  adjusted  to  make  all  the  guns  shoot  together? 

Note  that  no  one  of  the  guns  shoots  closely  enough  to  be  taken  as  a  standard  gun. 


12 
At  8500  yards,  one  yard  in  deflection  equals  -^^r 


:.15  knot  on  deflection  scale. 


Number 
of  gun. 

Errors. 

With  reference  to  point  P,  each 
gun  shot. 

To  bring  all  sights  to- 
gether  set  them  for 
each  gun  as  follows : 

Range. 
Yds. 

Deflection. 
Yds. 

In  range. 

Yds. 

In  deflection. 

In  range. 

Yds." 

In  deflec- 
tion. 
Kts. 

Yards. 

Knots. 

1 

25  short 

100  over 

100  short 

75  short 

150  over 

75  short 

80  short 

90  over 

25  left 
50  left 
75  right 
50  right 
100  left 
75  left 
80  right 
25  right 

25  short 

100  over 

100  short 

75  short 

150  over 

75  short 

80  short 

90  over 

25  left 
50  left 
75  right 
50  right 
100  left 
75  left 
80  right 
25  right 

3.75  left 

7.5    left 
11.25  right 

7.5    right 
15.0    left 
11.25  left 
12.0    right 

3.75  right 

8525 
8400 
8600 
8575 
8350 
8575 
8580 
8410 

53  75 

2 

57.50 

3 

38  75 

4 

42.50 

5 

65.00 

6 

61  25 

7 

38.00 

8 

46.25 

After  the  sights  have  been  set  as  indicated  above,  move  the  sight  scales  under  the 
pointers  until  the  latter  are  over  the  8500-yard  marks  in  range  and  over  the  50-kuot 
marks  in  deflection,  and  then  clamp  the  scales. 


APPEXDIX  B. 
THE  FARNSWORTH  GUN  ERROR  COMPUTER. 

PURPOSE  AND  USE. 

1.  This  instrument  was  devised  by  Midshipman  J.  S.  Famsworth,  U.  S.  N.,  class 
of  1915,  during  his  first  class  year  at  the  Naval  Academy. 

2.  It  is  intended  for  the  purpose  of  determining  quickly  and  accurately,  by 
mechanical  means  and  without  computations,  the  errors  in  range  and  in  deflection 
introduced  into  gun  fire  by :  * 

(a)  Wind. 

(b)  ]\Iotion  of  firing  ship. 

(c)  Motion  of  target  ship. 

(d)  Variation  from  standard  in  the  temperature  of  the  powder, 

(e)  Variation  from  standard  in  the  density  of  the  atmosphere. 

Plate  I  shows  the  device  on  an  enlarged  scale,  so  that  the  graduations  can  be 
clearly  seen.  The  radial  arm  shown  at  the  right  of  the  drawing  is  secured  to  the 
same  axis  as  the  discs. 

3.  The  uses  and  advantages  of  the  instrument  are  readily  apparent.  It  can  be 
used  by  both  spotting  and  plotting  groups  if  desired,  but  presumably  it  would  be  used 
in  the  plotting  room.  Its  use  will  enable  the  initial  errors  to  be  allowed  for  in  firing 
ranging  shots  to  be  accurately  and  quickly  determined,  so  that  with  it  a  spotter  has  a 
vastly  greater  chance  of  having  the  ranging  shot  strike  within  good  spotting  distance 
of  the  target  than  by  any  "  judgment "  or  "  rule  of  thumb  "  methods.  This  should 
enable  a  ship  to  begin  to  place  her  salvos  properly  in  a  shorter  time  and  with  less 
waste  of  ammunition  than  could  be  done  without  the  device. 

4.  Errors  due  to  changes  in  courses,  speeds,  wind,  or  other  conditions  during 
firing  can  be  similarly  quickly  obtained  by  the  use  of  the  computer. 

5.  The  accompanying  drawing  (Plate  I)  shows  the  device  arranged  for  working 
with  apparent  wind,  and  for  determining  deflection  errors  directly  in  knots  of  the 
deflection  scale  of  the  sight,  and  not  in  yards.  The  device  could  be  equally  well 
arranged  for  real  wind,  for  deflections  in  yards,  or  for  any  other  desired  system,  by 
simply  drawing  the  proper  spiral  curves  on  the  smaller  disc;  but  the  arrangement 
shown  here  is  believed  to  be  the  most  useful  one  for  service  conditions.    The  drawing 

*  Throughout  this  description  the  "  errors  "  have  been  considered  and  not  the  "  cor- 
rections." In  the  practical  use  of  the  computer  it  must  be  remembered  that,  having 
determined  an  "  error  "  tlie  "  correction  "  to  compensate  for  it  is  numerically  equal  but  of 
opposite  sign.    Thus,  an  "  error  "  of  100  short  calls  for  a  correction  of  "  up  100,"  etc. 

A  very  clear  and  concise  statement  of  the  purpose  and  principle  of  the  gun  error  com- 
puter is  contained  in  the  following  extract  from  a  report  thereon  submitted  to  the  com- 
mander-in-chief of  the  United  States  Atlantic  fleet  by  Ensign  H.  L.  Abbott,  U.  S.  N. : 

"  The  gun  error  computer  is  a  combination  of  a  set  of  curves  showing  the  correction  to 
be  applied  at  various  ranges  to  range  and  deflection  for  unit  variation  from  normal  of  the 
conditions  considered,  such  as  wind ;  and  of  a  specially  graduated  numerical  or  circular 
slide  rule  for  modifying  the  correction  for  unit  variation  to  give  the  correction  for  the 
actual  variation.  This  instrument  can  be  made  to  take  the  place  of  the  range  tables,  and 
with  its  aid  the  corrections  for  any  particular  set  of  conditions  can  be  picked  out  with 
much  greater  ease  and  facility  than  with  the  present  cumbrous  range  tables  and  accom- 
panying necessary  calculations." 


316  APPENDICES 

does  not  show  the  three  curves  in  colors,  as  they  should  be  drawn  on  a  working  device, 
each  curve  being  of  a  radically  different  color  from  the  others;  and  the  powder 
temperature  error  and  density  error  curves  are  not  shown  on  the  drawing.  In  the 
following  descriptions  it  is  assumed  that  the  several  cui-ves  would  be  drawn  as  follows : 

Wind  range  curve  in  red. 

Wind  deflection  curve  in  green. 

Target  and  gun  range  curve  in  black. 

Powder  temperature  error  curve  in  blue. 

Density  error  curve  in  yellow. 

METHOD  OF  CONSTRUCTION. 

6.  In  external  appearance  and  in  some  principles  of  construction,  the  device  is 
similar  to  an  omnimeter.  It  consists,  as  shown  on  the  plate,  of  two  circular  discs, 
an  outer  or  larger,  and  an  inner  or  smaller  one,  concentrically  secured  on  the 
same  axis  and  capable  of  independent  rotary  motion  around  that  axis;  and,  in 
addition,  of  a  radial  arm  secured  on  the  same  axis  and  capable  of  free  rotary  motion 
around  that  axis.  These  parts  should  be  so  arranged  that  the  radial  arm  can  be 
clamped  to  the  inner  disc  without  clamping  the  two  discs  together,  and  so  that  the 
two  discs  can  be  clamped  together  without  clamping  the  radial  arm  to  the  inner 
disc.  The  radial  arm  should  be  made  of  some  transparent  material,  with  the  range 
scale  line  scribed  radially  from  the  center  of  the  axis  down  the  middle  of  the  arm. 

7.  The  salient  features  of  the  device  are : 

(a)  The  Range  Circle. — The  graduations  on  the  outside  of  the  larger  circle  on 
the  larger  disc. 

(b)  The  Deflection  Circle. — The  graduations  on  the  inside  of  the  larger  circle 
on  the  larger  disc. 

(c)  The  Speed  Circle. — The  graduations  on  the  outside  of  the  smaller  circle  on 
the  larger  disc.  This  circle  is  in  coincidence  with  the  periphery  of  the  smaller  or 
inner  disc. 

(d)  The  Correction  Circle. — The  graduations  on  the  periphery  of  the  smaller 
disc. 

(e)  The  Range  and  Deflection  Curves. — Drawn  on  the  face  of  the  smaller  disc, 
spirally,  from  the  center  of  the  disc  outward.    They  are  the : 

(1)  Wind  range  curve. 

(2)  Wind  deflection  curve. 

(3)  Target  and  gun  range  curve. 

(4)  Powder  temperature  curve. 

(5)  Density  curve. 

(f )  The  Radial  Arm. — Bearing  the  range  scale. 

Of  the  above,  a,  b  and  c  are  all  on  the  larger  disc,  and  their  positions  relative 
to  one  another  are  therefore  fixed.  Also  d  and  e  are  both  on  the  smaller  disc,  and 
their  positions  relative  to  each  other  are  therefore  fixed.  However,  a,  h  and  c  can  be 
rotated  relative  to  d  and  e.  The  range  scale,  being  on  /_,  can  be  rotated  relative  to 
either  or  to  both  of  the  discs. 

8.  Of  the  above,  only  the  curves  vary  for  different  guns.  It  would  therefore  be 
necessary  to  construct  the  apparatus  and  then  have  the  curves  scribed  on  it  for  the 
particular  type  of  gun  with  which  the  individual  instrument  is  to  be  used.  Thus, 
there  would  be  one  computer  for  each  caliber  of  gun  on  board.  Plate  I  shows  the 
device  as  arranged  for  the  12"  gun  for  which  7  =  2900  f.  s.,  w  =  870  pounds  and 
c  =  0.61;  the  necessary  data  for  its  construction  having  been  obtained  from  the  range  . 
table  for  that  srun. 


APPENDICES  317 

9.  The  mathematical  principles  involved  in  the  construction  of  the  several  ele- 
ments of  the  device  are  described  herein  (the  description  being  based  on  the  assump- 
tion that  the  reader  is  not  familiar  with  the  omnimeter). 

(a)  Range  Circle. — The  entire  circumference  is  divided  into  parts  representing 
logarithmic  increments  in  the  secant  of  the  angle,  from  zero  degrees  to  the  angle 
whose  logarithmic  secant  is  unity  (84°  + ).  These  increments  are  laid  down  on  the 
circle  in  a  counter-clockwise  direction  according  to  the  logarithmic  secants,  and  the 
scale  is  marked  with  the  angles  corresponding  to  the  given  logarithmic  secants.  For 
example,  the  point  marked  23°  lies  in  a  counter-clockwise  direction  from  the  zero  of 
the  scale,  and  at  a  distance  from  it  equal  to  .03597  of  the  circumference  (log  sec  23° 
=  0.03597). 

(b)  Deflection  Circle. — The  entire  circumference  is  divided  into  parts  repre- 
senting logarithmic  increments  in  the  sine  of  the  angle,  from  the  angle  whose 
logarithmic  sine  is  9.00000  —  10(5°  +  )  to  90°.  These  increments  are  laid  down  on 
the  circle  in  a  clockwise  direction  according  to  the  logarithmic  sines,  and  the  scale 
is  marked  with  the  angles  corresponding  to  the  given  logarithmic  sines.  For 
example,  the  point  marked  23°  lies  in  a  clockwise  direction  from  the  zero  of  the  scale, 
and  at  a  distance  from  it  equal  to  .59188  of  the  circumference  (log  sine  23° 
=  9.59188  —  10).  The  zero  of  the  scale  coincides  with  the  zero  of  the  scale  of  the 
range  circle. 

(c)  Speed  Circle. — The  entire  circumference  is  divided  into  parts  representing 
logarithmic  increments  in  the  natural  numbers  from  1.0  to  10  (the  decimal  point 
may  be  placed  wherever  necessary,  and  the  point  marked  "  10  "  may  be  considered  as 
the  "  zero  "  of  this  scale,  and  will  hereafter  be  referred  to  as  such  in  this  description). 
The  increments  are  laid  down  on  the  circle  in  a  counter-clockwise  direction  from 
zero,  and  the  divisions  of  the  scale  are  marked  with  the  numbers  corresponding  to 
the  given  logarithms.  For  example,  the  number  2.3  lies  in  a  counter-clockwise  direc- 
tion from  the  zero,  and  at  a  distance  from  it  equal  to  .36173  of  the  circumference 
(log  2.3  =  0.36173).  The  zero  of  this  scale  coincides  with  the  zeros  of  the  range  and 
deflection  circles. 

(d)  Correction  Circle. — The  construction  of  the  correction  circle  is  exactly  the 
same  as  that  of  the  speed  circle,  except  that  the  scale  is  laid  down  in  a  clockwise 
direction  from  the  zero.''' 

(e)  Range  and  Deflection  Curves. — Each  of  these  is  based  on  the  data  in  the 
appropriate  column  of  the  range  table  for  the  given  gim,  and  these  curves  are  there- 
fore different  for  different  guns.    The  method  of  plotting  them  is  described  below. 

(f )  Range  Scale. — A  radius  of  appropriate  length  to  fit  the  discs  is  subdivided 
as  a  range  scale.  These  divisions  are  purely  arbitrary,  and  Plate  I  shows  the  increments 
in  range  as  decreasing  in  relative  magnitude  on  the  scale  as  the  range  increases  ;  so  that 
the  divisions  are  larger  and  more  easily  and  accurately  read  at  the  ranges  that  will 
most  likely  be  used ;  becoming  smaller  as  the  range  becomes  very  great.  The  size  of 
these  divisions,  either  actual  or  relative  to  one  another,  does  not  affect  the  work  of  the 
instrument,  provided  this  range  scale  be  prepared  first  and  then  used  in  plotting  the 
error  curves  in  the  manner  described  below. f 

*  Those  familiar  with  the  omnimeter  will  perceive  that  up  to  this  point  the  prin- 
ciples of  that  instrument  have  been  followed:  but  that  the  scales  of  the  range  circle 
(logarithmic  secants)  and  speed  circle  (logarithms  of  numbers)  have  been  laid  down  in 
the  opposite  direction  from  those  on  the  omnimeter. 

f  If  the  device  be  made  of  a  good  working  size,  these  divisions  may  all  be  made  of 
the    same    size    and    still   be    clearly    read,    and    this    is    the    best   way    to    construct    it. 


318  APPENDICES 

10.  To  plot  the  range  and  deflection  curves  for  wind  and  for  motion  of  firing  or 
target  ship,  the  data  is  obtained  from  the  proper  column  in  the  range  table  for  the 
error  and  gun  under  consideration  (Columns  13,  14,  15,  16,  17  and  18).  Thus,  for 
instance,  to  locate  the  point  of  the  wind  range  curve  for  a  range  of  10,000  yards  for 
the  given  13"  gun.  Column  13  of  the  range  table  (Bureau  Ordnance  Pamphlet  No. 
298)  shows  that  the  error  in  range  caused  by  a  12-knot  wind  blowing  along  the  line  of 
fire  is  27  yards,  and  it  would  therefore  be  2.25  yards  for  a  1-knot  wind.  Therefore 
the  desired  point  of  the  curve  is  plotted  on  the  inner  disc  on  a  radius  passing  through 
the  2.25  mark  on  the  correction  circle,  and  at  a  distance  from  the  center  correspond- 
ing to  10,000  yards  on  the  range  scale.  Enough  points  are  plotted  in  this  manner  to 
enable  an  accurate  curve  to  be  scribed  through  them.  The  other  curves  are  plotted 
in  a  similar  manner,  but  instead  of  plotting  deflection  curves  in  "  yards  error,"  they 
are  plotted  in  "  knots  error  of  the  deflection  scale  of  the  sight,"  thus  enabling  the 
deflection  error  to  be  determined  directly  in  knots  for  application  to  the  sight  drums. 
For  example,  for  the  wind  deflection  curve,  the  data  for  plotting  would  be  found  by 
dividing  the  data  in  Column  16  of  the  range  table  for  the  given  range  by  the  corre- 
sponding data  in  Column  18.  Approximate  values  of  the  correction  scale  reading 
are  marked  at  intervals  along  the  curves  to  aid  the  operator  in  placing  the  decimal 
point  correctly  and  in  getting  the  result  in  correct  units. 

11.  To  plot  the  powder  temperature  curve,  it  will  be  seen  from  Column  10  of 
the  range  table  that,  for  a  range  of  15,000  yards,  50  f.  s.  variation  from  standard  in 
the  initial  velocity  causes  377  yards  error  in  range,  therefore  1  foot-second  variation 
in  initial  velocity  would  cause  7.54  yards  error  in  range.  From  the  notes  to  the 
range  table,  it  will  be  seen  that,  for  this  gun,  10°  variation  from  standard  in  the 
temperature  of  the  powder  (90°  being  the  standard)  causes  a  change  of  35  f.  s.  in 
initial  velocity,  therefore  a  variation  of  one  degree  in  temperature  would  cause  a 
change  of  3.5  f.  s.  in  initial  velocity.  Consequently,  a  variation  of  one  degree  from 
standard  in  the  temperature  of  the  powder  would  cause  an  error  of  3.5  x  7.54  =  26.39 
yards  in  range.  Therefore,  to  locate  the  point  on  the  curve  corresponding  to  15,000 
yards  range,  place  the  desired  point  on  the  face  of  the  smaller  disc  on  a  radius  passing 
through  the  26.39  mark  on  the  correction  circle,  and  at  a  distance  from  the  center 
corresponding  to  15,000  yards  on  the  range  scale. 

12.  Before  proceeding  to  a  description  of  the  method  of  plotting  the  density 
curve,  a  brief  preliminary  discussion  of  another  point  is  necessary.  As  the  density 
of  the  air  depends  upon  two  different  variables,  one  being  the  barometric  reading  and 
the  other  the  temperature  of  the  air  (assuming,  as  is  done  in  present  methods,  that 
the  air  is  always  half  saturated),  it  is  not  practicable  from  a  point  of  view  of  easy 
operation  to  lay  down  a  single  curve  for  use  in  determining  the  density  corresponding 
to  given  readings  of  barometer  and  thermometer.  Therefore  a  sheet  of  auxiliary 
curves  is  necessary  for  this  purpose,  for  use  in  connection  with  the  computer.  Such 
a  set  of  curves  is  shown  on  Plate  II,  and  is  good  for  any  gun.  It  is  really  a  graphic 
representation  of  Table  IV  of  the  Ballistic  Tables  (the  table  of  multipliers  for 
Column  12),  and  values  of  the  multiplier  can  be  taken  from  these  curves  much  more 
quickly  than  they  can  be  obtained  from  the  table  by  interpolation.  These  curves 
have  been  designated  atmospheric  condition  curves,  and  on  Plate  II  show  as  straight 
lines,  giving  values  for  the  multipliers  ten  times  as  great  as  those  given  in  the 
table.  This  has  been  done  in  order  to  have  the  computer  retain  the  principle  on 
which  it  is  constructed  for  all  other  errors;  namely,  that  the  first  reading  taken  from 
the  correction  circle  by  bringing  the  range  mark  on  the  range  scale  into  coincidence 
with  the  proper  curve  shall  show  the  error  due  to  unit  variation  in  the  quantity  under 
consideration.     (The  same  thing  could  be  done  iu  this  case  by  plotting  the  curve  for 


APPENDICES  319 

the  full  errors  due  to  10  per  cent  variation  in  the  density,  and  then  using  the  values 
of  the  multipliers  as  given  in  the  table;  but  this  would  make  the  principle  of  con- 
struction different  in  this  case  from  what  it  is  in  all  the  others,  and  it  was  deemed  best 
to  adhere  to  the  same  principle  throughout.)  It  will  also  be  seen  that  the  curves  in 
question  are  plotted  as  straight  lines,  whereas  they  would  not  be  quite  that  if 
accurately  plotted  from  the  table.  The  straight  lines  have  been  plotted  as  repre- 
senting as  nearly  as  possible  the  mean  value  of  the  curve  that  would  be  obtained  by 
plotting  accurately  from  the  table.  At  the  end  of  this  description  will  be  found  a 
mathematical  demonstration  of  the  fact  tliat  these  curves  must  be  straight  lines, 
whence  it  follows  that  the  table  is  theoretically  slightly  in  error  in  so  far  as  it  departs 
from  this  requirement. 

13.  To  plot  the  density  curve,  it  will  be  seen  from  Column  12  of  the  range  table 
that,  for  a  range  of  15,000  yards,  a  variation  from  standard  in  the  density  of  the 
atmosphere  of  10  per  cent  will  cause  an  error  in  range  of  451  yards;  therefore,  a 
variation  of  1  per  cent  in  density  will  cause  an  error  of  45.1  yards.  Therefore  to 
plot  the  point  on  the  curve  corresponding  to  15,000  yards  range,  place  the  desired 
point  on  the  face  of  the  disc  on  a  radius  passing  through  the  45.1  mark  on  the  cor- 
rection circle,  and  at  a  distance  from  the  center  corresponding  to  the  15,000-yard 
mark  on  the  range  scale.  Proceed  in  a  similar  manner  to  plot  points  corresponding 
to  enough  other  ranges  to  enable  the  curve  to  be  accurately  scribed. 

METHOD  OF  USE. 

14.  Before  proceeding  to  describe  the  use  of  the  computer,  the  following  rule 
must  be  laid  down  : 

(a)  Always  use  the  angle  that  is  less  than  90°  that  any  of  the  directions  makes 
with  the  line  of  fire,  in  order  that  we  may  always : 

(b)  Determine  all  range  errors  involving  angles  by  multiplying  results  in  the 
line  of  fire  by  the  cosine  of  the  angle;  that  is,  by  dividing  by  the  secant. 

(c)  Determine  all  deflection  errors  involving  angles  by  multiplying  results 
perpendicular  to  the  line  of  fire  by  the  sine  of  the  angle. 

15.  As  an  illustration  of  the  method  of  using  the  computer,  its  manipulation 
will  now  be  described  in  finding  the  error  in  range  that  would  be  caused  by  an 
apparent  wind  of  45  knots  an  hour  blowing  at  an  angle  of  50°  to  the  line  of  fire,  for  a 
range  of  15,000  yards.    The  gun  is  the  same  13"  gun. 

16.  Move  the  radial  arm  until  the  15,000-yard  mark  on  the  range  scale  cuts  the 
wind  range  curve  (red  curve).  The  value  on  the  correction  circle  where  it  is  now  cut 
by  the  range  scale  line  is  the  error  for  1-knot  wind  in  the  line  of  fire,  and  will  show 
as  5  yards.  Now  swing  the  inner  disc  and  radial  arm  together  until  the  range  scale 
line  and  5  on  the  correction  circle  are  in  coincidence  with  the  45  mark  of  the  speed 
circle  and  clamp  the  two  discs  together.  The  reading  now  showing  on  the  correction 
circle  in  coincidence  with  the  zero  of  the  speed  circle  is  235  yards,  or  the  product  of 
5  X  45,  and  this  would  be  the  error  caused  by  a  45-knot  wind  blowing  along  the  line 
of  fire.  This  is  not  what  is  wanted  in  this  case,  however,  so  with  the  two  discs  still 
clamped  together,  swing  the  radial  arm  until  the  range  scale  line  is  in  coincidence 
with  the  50°  mark  on  the  range  circle,  and  then  read  across  by  the  range  scale  line 
to  the  correction  circle,  where  the  coincident  mark  will  be  144  yards,  which  is  the 
desired  result,  and  will  be  found  to  check  with  the  results  of  work  with  the  range 
table  by  ordinary  methods. 

17.  The  above  process  may  be  more  fully  explained  as  follows,  for  the  benefit  of 
those  who  are  not  familiar  with  the  omnimeter.  As  a  result  of  the  manner  in  which 
the  wind  range  curve  (red)  was  plotted,  when  the  15,000-yard  mark  on  the  range 


320  APPENDICES 

scale  was  brought  to  cut  the  wind  range  curve,  and  the  reading  was  noted  where  the 
range  scale  line  cut  the  correction  circle,  that  reading  was  5  yards,  or  the  error  due  to 
a  1-knot  wind  blowing  along  the  line  of  fire.  JSTow  had  the  zeros  of  the  correction  and 
speed  circles  been  in  coincidence  when  this  was  done,  when  the  inner  disc  was  moved 
around  in  a  counter-clockwise  direction  until  the  five  of  the  correction  circle  coin- 
cided with  the  zero  of  the  speed  circle,  the  zero  of  the  correction  circle  moved  a  dis- 
tance equal  to  log  5.  Now  when  the  motion  of  the  inner  disc  was  continued  in  the 
same  direction  until  the  5  of  the  correction  circle  coincided  with  the  45  of  the  speed 
circle,  the  zero  of  the  correction  circle  moved  a  further  distance  in  the  same  direction 
equal  to  log  45.  The  total  travel  of  the  zero  of  the  correction  circle  must  therefore 
have  been  log  5  4- log  45  =  log  225 ;  and  the  reading  on  the  speed  circle  now  coincident 
with  the  zero  of  the  correction  circle  (the  measure  of  the  total  travel  of  that  zero) 
must  be  225  yards,  which  is  the  error  in  range  due  to  a  45-knot  wind  in  the  line  of 
fire.  The  decimal  point  has  moved  one  digit  to  the  right  because  of  the  fact  that  the 
zero  of  the  correction  circle  traveled  between  one  and  two  complete  circumferences 
during  the  operation.  Now  clamp  the  two  discs  together  as  they  stand.  If  the  range 
scale  on  the  radial  arm  be  first  placed  at  the  zero  of  the  speed  circle  (where  we  read 
225  on  the  correction  circle),  which  is  also  the  zero  of  the  range  or  log  secant  circle; 
and  then  be  moved  in  a  counter-clockwise  direction  until  the  range  scale  line  is  coin- 
cident with  the  50°  mark  on  the  range  circle,  the  range  scale  line  will  have  traveled 
a  distance  from  the  225  mark  on  the  correction  circle  equal  to  log  sec  50°,  and  if  the 
range  scale  line  be  then  followed  across  from  the  50°  mark  on  the  range  circle  to  the 
correction  circle,  the  reading  on  the  latter  will  be  log  225  — log  sec  50°,  or  log  225 
+  logcos50°;  that  is,  the  logarithm  of  225  X  cos  50°,  which  is  the  desired  result; 
and  reading  off  the  anti-logarithm  on  the  correction  circle  corresponding  to  the  above 
result,  the  reading  will  be  144  yards,  which  is  the  desired  error ;  that  is,  the  error  in 
range  caused  by  an  apparent  wind  of  45  knots  blowing  at  an  angle  of  50°  with  the 
line  of  fire.  The  sign  of  the  error,  that  is,  whether  it  is  a  "  short "  or  an  "  over," 
will  at  once  be  apparent  from  a  glance  at  the  plotting  board,  on  which  the  direction 
of  the  apparent  wind  should  be  indicated  relative  to  the  line  of  fire. 

18.  To  determine  the  deflection  due  to  the  wind,  proceed  in  a  similar  manner, 
using  the  wind  deflection  curve  (green),  and  taking  the  angle  from  the  deflection 
circle.  Setting  the  radial  arm  with  the  15,000-yard  mark  of  the  range  scale  in 
coincidence  with  the  wind  deflection  curve  gives,  from  the  correction  circle,  that  a 
1-knot  wind  perpendicular  to  the  line  of  flre  causes  an  error  of  0.245  knots  (on  the 
deflection  scale  of  the  sight)  in  deflection.  Moving  the  .245  mark  of  the  correction 
circle  around  to  coincide  with  the  45-knot  mark  on  the  speed  circle  and  reading  the 
zero  of  the  correction  (or  speed)  circle  will  give  11.0  knots  as  the  error  due  to  a 
45-knot  wind  perpendicular  to  the  line  of  flre  (this  would  not  be  noted  in  actual 
practice  unless  the  wind  were  actually  blowing  perpendicularly  to  the  line  of  fire,  in 
which  case  it  would  be  the  desired  result)  ;  and  reading  across  from  the  50°  mark  on 
the  deflection  circle  to  the  correction  circle  would  give  8.5  knots  as  the  amount  of 
error  in  deflection.  As  before,  the  sign  of  the  error  must  be  determined  from  the 
plotting  board.  What  was  really  done  here,  after  determining  the  value  0.245,  was 
to  perform  the  addition  log  0.245 -flog  45  =  log  11,  and  then  the  addition  log  11 
-flog  sin  50°=  log  8.5.  That  is,  the  value  0.245  was  first  found  mechanically,  and 
then  the  compound  operation  0.245  x  45  x  sin  50°  =  8.5  was  mechanically  performed. 

19.  As  the  apparent  wind  was  used  in  the  preceding  operations,  the  errors  for 
the  motion  of  the  gun  would  be  taken  from  the  same  curve  as  those  for  the  motion 
of  the  target.  For  the  error  in  range  the  method  is  exactly  the  same  for  both  gun 
motion  and  target  motion  as  for  the  wind  error  in  range,  using  the  target  and  gun 


APPENDICES  321 

range  curve  (black).    For  the  error  in  deflection  the  work  might  be  done  in  either 
one  of  two  ways,  as  follows : 

(a)  Use  the  target  and  gun  range  curve  (black)  for  deflection  errors  as  well  as 
for  range  errors  (Columns  15  and  18  of  the  range  table  are  numerically  the  same) 
proceeding  as  before,  which  would  give  the  deflection  error  in  yards,  which  would 
then  have  to  be  transformed  into  knots  of  the  sight  deflection  scale. 

(b)  Solve  by  the  principles  laid  down  in  paragraph  24,  subparagraph  (b),  below, 
for  the  solution  of  right  triangles.  This  can  be  done  because  what  we  desire  is  the 
resolution  of  the  speed  into  a  line  at  right  angles  to  the  line  of  fire,  which  is  the  speed 
in  knots  multiplied  by  the  sine  of  the  angle,  the  result  being  in  knots  of  the  deflection 
scale.  This  is  the  shortest  method,  requires  no  curve  on  the  computer,  and  is  the 
one  actually  used  in  practice.  Suppose  the  firing  ship  were  steaming  at  15  knots  on  a 
course  making  an  angle  of  36°  with  the  line  of  fire.  Bring  the  15  on  the  correction 
circle  into  coincidence  with  the  zero  of  the  speed  circle.  Then  read  across  from  the 
36°  mark  on  the  deflection  circle  to  the  correction  circle,  where  the  8.8  mark  shows 
that  the  required  error  is  8.8  knots  of  the  deflection  scale  of  the  sight.  The  operation 
here  performed  was  15  X  sin  36°. 

20.  For  the  error  in  range  due  to  the  motion  of  the  target,  proceed  exactly  as  was 
done  in  the  case  of  motion  of  the  gun,  using  the  same  curve ;  the  target  and  gun 
range  curve  (black).  The  process  for  the  deflection  error  is  also  the  same  as  before. 
Suppose  the  target  were  moving  at  18  knots  at  an  angle  of  40°  with  the  line  of  fire. 
Put  18  on  the  correction  circle  in  coincidence  with  the  zero  of  the  speed  circle.  Then 
read  across  from  40°  on  the  deflection  circle  to  the  correction  circle,  and  11.6  knots 
will  there  be  shown  as  the  required  error. 

21.  To  use  the  powder  temperature  curve,  bring  the  range  scale  into  coincidence 
with  the  powder  temperature  curve  at  the  given  range  mark,  and  clamp  the  radial 
arm  and  smaller  disc  together.  Determine  the  variation  from  standard  (90°  F.) 
in  the  temperature  of  the  powder  (90° '^t°  =  T° ;  where  ^°  is  the  temperature  of  the 
charge,  and  T°  is  the  variation  from  standard).  Turn  the  smaller  disc  and  radial 
arm  together  until  the  range  scale  line  cuts  the  speed  circle  at  the  T°  mark.  Then 
read  the  mark  on  the  correction  circle  that  is  coincident  with  the  zero  mark  on  the 
speed  circle  (or  the  mark  on  the  speed  circle  that  is  coincident  with  the  zero  mark 
on  the  correction  circle),  and  this  reading  will  be  the  desired  error  in  yards  resulting 
from  a  variation  of  T°  from  standard  (90°  F.)  in  the  temperature  of  the  powder. 
A  powder  temperature  higher  than  standard  always  gives  an  increase  in  range,  and 
the  reverse. 

22.  To  use  the  density  curve,  bring  the  range  scale  into  coincidence  with  the 
density  error  curve  for  the  given  range,  and  clamp  the  radial  arm  and  smaller  disc 
together.  Determine  the  value  of  the  multiplier  for  the  given  barometer  and 
thermometer  readings  from  the  atmospheric  condition  curves.  Turn  the  smaller  disc 
and  radial  arm  together  until  the  range  scale  line  cuts  the  speed  circle  at  the  mark 
indicating  the  value  of  the  multiplier  thus  determined.  Then  read  the  mark  on  the 
correction  circle  that  is  coincident  with  the  zero  mark  on  the  speed  circle  (or  the 
mark  on  the  speed  circle  that  is  coincident  with  the  zero  of  the  correction  circle), 
and  this  reading  will  be  the  desired  error  m  yards  resulting  from  the  variation  from 
standard  in  the  density  of  the  atmosphere  due  to  the  given  barometric  and  ther- 
mometric  readings.  A  multiplier  carrying  a  negative  sign  (that  is,  one  taken  from 
a  red  point  on  the  atmospheric  condition  curves)  always  gives  a  "short"  (density 
greater  than  unity)  ;  and  one  carrying  a  positive  sign  (that  is,  one  taken  from  a 
black  point  on  the  atmos^jheric  condition  curves)  always  means  an  "  over  "  (density 
less  than  unity). 

21 


323 


APPENDICES 


23.  Having  shown  how  to  manipulate  the  computer  in  detail,  it  will  be  seen 
that  the  process  of  use  in  the  plotting  room  would  be  about  as  follows : 


FORM   FOR  USE   IN   CONNECTION  WITH    FARNSWORTH   ERROR   COMPUTER. 


Range, 


yards. 


Temperature  of  powder: 

Standard,  90° ;   actual,  — °. 

Atmospheric  conditions: 

Barometer,  — ";  tlier.,  — ". 

Motion  of  gun: 

Speed,  —  knots;   angle, — °. 

Apparent  wind: 

Velocity,  —  knots;  angle,  — ° 

Sums. 

Preliminary  errors. 

Motion  of  target: 

Speed,  —  knots;   angle, — °. 

Final  errors  (signs  to  be  changed 
to  give  "  corrections  ") . 


Errors  in. 


Range. 
Yds. 


Short. 


Over. 


Deflection. 
Knots. 


Right. 


Left. 


•(a)  By  "angle"  is  meant  that  angle  le.^s  than  90°  which  the  course  of  the 
firing  ship,  direction  of  the  apparent  wind,  or  course  of  the  target  ship  makes  with 
the  line  of  fire. 

(b)  The  preliminary  errors  include  all  those  that  will  presumably  be  known 
long  enough  in  advance  to  afford  reasonable  time  for  their  determination. 

(c)  The  temperature  of  the  powder  and  the  readings  of  the  barometer  and 
thermometer  will  be  known  before  starting  the  approach.  The  first  two  lines  of  the 
above  form  may  therefore  be  filled  out  when  work  begins,  and  will  presumably 
remain  constant  throughout  the  action. 

(d)  As  soon  as  plotting  begins  and  the  proposed  line  of  fire  and  range  are 
determined  with  sufficient  accuracy,  the  plotter  determines  the  angles  made  by  the 
course  of  the  firing  ship  and  by  the  apparent  wind  (the  information  relative  to  the 
latter  being  sent  down  by  the  spotter)  with  the  proposed  line  of  fire,  and  the  errors 
for  gun  motion  and  wind  are  determined  and  entered  in  their  proper  columns.  The 
algebraic  additions  necessary  to  give  the  preliminary  errors  are  then  made  and 
entered.  This  leaves  only  target'  motion  to  be  accounted  for,  and  as  soon  as  the 
plotter  has  the  necessary  information  he  gives  the  "  angle  "  and  speed  of  the  target 
ship,  the  errors  caused  thereby  are  taken  from  the  computer  and  entered  in  their 
columns,  and  then  two  simple  algebraic  additions  give  the  total  errors  required. 
The  necessary  corrections  for  application  to  the  sights  for  the  ranging  shot  can  then 
be  sent  to  the  guns. 

24.  The  computer  is  readily  available  for  the  solution  of  any  right  triangle,  in 
addition  to  the  purpose  for  which  it  was  devised.  In  the  case  of  angle  from  84°  to 
9U°,  the  sines  are  practically  equal  to  unity  and  the  cosines  are  negligible,  and 


APPENDICES  323 

oppositely  for  angles  from  0°  to  6°.  For  this  reason  the  graduations  for  these  angles 
have  been  omitted  from  the  circles.  For  examples  in  the  solution  of  right  triangles 
we  have : 

(a)  Given  One  Angle  and  the  Hypothenuse  to  Find  the  Side  Adjacent. — Given 
that  the  angle  is  30°  and  the  hypothenuse  is  27.  Put  27  on  the  correction  circle 
in  coincidence  with  zero  on  the  speed  circle.  Then  find  30°  on  the  range  circle  and 
read  across  to  the  correction  circle,  where  23.5  will  be  found  for  the  side  adjacent. 
(27  X  cos  30°  =27  divided  by  sec  30°  =23.4  by  the  traverse  tables.) 

(b)  Given  One  Angle  and  the  Hypothenuse  to  Find  the  Side  Opposite. — Given 
that  the  angle  is  30°  and  the  hypothenuse  is  27.  Put  27  on  the  correction  circle  in 
coincidence  with  zero  on  the  speed  circle.  Then  find  30°  on  the  deflection  circle  and 
read  across  to  the  correction  circle,  where  13.5  will  be  found  for  the  side  opposite. 
(27  X  sin  30°  =  13.5  by  the  traverse  tables.) 

(c)  Given  One  Angle  and  the  Side  Opposite  to  Find  the  Hypothenuse. — Given 
that  the  angle  is  30°  and  the  side  opposite  is  15.  Put  15  on  the  correction  circle  in 
coincidence  with  30°  on  the  deflection  circle,  and  coinciding  with  the  zero  of  the 
correction  circle  will  be  30+  on  the  speed  circle  for  the  hypothenuse.  (15  x  cosec  30° 
=  15  divided  by  sin  30°  =  30+  by  the  traverse  tables.) 

(d)  Given  One  Angle  and  the  Side  Adjacent  to  Find  the  Hypothenuse. — Given 
that  the  angle  is  30°  and  the  side  adjacent  is  15.  Put  15  on  the  correction  circle  in 
coincidence  with  30°  on  the  range  circle,  and  coinciding  with  the  zero  of  the  correc- 
tion circle  will  be  17.25  on  the  speed  circle  for  the  hypothenuse.  (15  x  sec  30°  =  17.3 
by  the  traverse  tables.) 

(e)  Given  One  Angle  and  the  Side  Adjacent  to  Find  the  Side  Opposite. — Given 
the  side  adjacent  as  15  and  the  angle  as  30°,  first  find  the  hypothenuse  as  in  (d), 
which  is  17.25.  Put  17.25  on  the  correction  circle  in  coincidence  with  zero  on  the 
speed  circle,  and  in  coincidence  with  30°  on  the  deflection  circle  will  be  found  8.G5 
on  the  correction  circle  for  the  side  opposite.  (17.25  x  sin  30°  =8.62  by  the  traverse 
tables.) 

(f)  Given  One  Angle  and  the  Side  Opposite  to  Find  the  Side  Adjacent. — Given 
the  side  opposite  as  15  and  the  angle  as  30°,  first  find  the  hypothenuse  as  in  (c), 
which  is  30  +  .  Bring  30+  on  the  correction  circle  into  coincidence  with  zero  on 
the  speed  circle,  and  in  coincidence  with  30°  on  the  range  circle  will  be  26+  on  the 
correction  circle  for  the  side  adjacent.  (30+ X  cos  30°  =  30  + divided  by  sec  30° 
=  26+  by  traverse  tables.) 

(g)  Given  the  Two  Sides  to  Find  the  Angles  and  Hypothenuse. — The  com- 
puter does  not  handle  this  ease  as  easily  as  the  traverse  tables  ;  but  it  is  not  one  usually 
encountered  in  the  class  of  work  wliere  the  instrument  would  habitually  be  used. 

INSTETJCTIONS  FOR  USE. 

25.  To  summarize,  the  following  brief  instructions  should  be  used  in  connection 
with  the  instrument : 

(a)  To  Determine  the  Error  in  Range  Resulting  from  a  Variation  from  Stand- 
ard in  the  Temperature  of  the  Powder. 

Error  in  Range. —  (Use  blue  curve.  Column  10  of  range  table.) 

(1)  Bring  the  given  range  on  the  range  scale  into  coincidence  with  the  powder 
temperature  curve,  and  clamp  the  radial  arm  and  smaller  disc  together. 

(2)  Determine  the  variation  from  standard  (90°  F.)  in  the  temperature  of  the 
powder  {20°  ^t°  =  T°). 


324  APPENDICES 

(3)  Turn  the  smaller  disc  and  radial  arm  together  until  the  range  scale  line 
cuts  the  speed  circle  at  the  T°  mark.  Eead  the  desired  error  on  the  correction  circle 
coincident  with  the  zero  of  the  speed  circle. 

(4)  A  powder  temperature  higher  than  the  standard  always  gives  an  increase  in 
range,  and  the  reverse. 

(b)  To  Determine  the  Error  in  Range  Resulting  from  a  Variation  from  Stand- 
ard in  the  Density  of  the  Atmosphere. 

Error  in  Range. —  (Use  yellow  curve,  Column  12.) 

(1)  Bring  the  given  range  on  the  range  scale  into  coincidence  with  the  density 
curve,  and  clamp  the  radial  arm  and  smaller  disc  together. 

(2)  Determine  the  value  of  the  multiplier  from  the  atmospheric  condition 
curves  for  the  given  readings  of  barometer  and  thermometer. 

(3)  Turn  the  smaller  disc  and  radial  arm  together  until  the  range  scale  line  cuts 
the  speed  circle  at  the  mark  indicating  the  value  of  the  multiplier  determined  from 
the  atmospheric  condition  curves.  Eead  the  desired  error  on  the  correction  circle 
coincident  with  the  zero  of  the  speed  circle. 

(4)  A  negative  sign  on  the  multiplier  alwa3's  means  a  "  short,"  and  a  positive 
sign  an  "  over." 

(c)  To  Determine  Errors  Due  to  an  Apparent  Wind  of  Known  Velocity  and  at 
a  Known  Angle  to  the  Line  of  Fire. 

Error  in  Range. —  (Use  red  curve,  Column  13.) 

(1)  Eotate  radial  arm  until  wind  range  curve  intersects  range  scale  on  runner 
at  given  range,  and  clamp  radial  arm  to  upper  disc. 

(2)  Eotate  lower  disc  until  range  scale  line  on  radial  arm  intersects  speed  circle 
at  apparent  wind  velocity  in  knots.    Clamp  discs  together;  unclamp  radial  arm. 

(3)  Eotate  radial  arm  until  range  scale  line  intersects  range  circle  at  angle  to 
line  of  fire  at  which  wind  is  blowing. 

(4)  Eead  across  by  range  scale  line  to  correction  circle,  and  note  result;  the 
desired  range  error  in  yards. 

(5)  Determine  sign  of  error  by  glance  at  plotting  board. 
Error  in  Deflection.     (Use  green  curve,.  Column  16.) 

(1)  Eotate  radial  arm  until  wind  deflection  curve  intersects  range  scale  on 
radial  arm  at  given  range,  and  clamp  radial  arm  to  inner  disc. 

(2)  Eotate  lower  disc  until  range  scale  line  on  radial  arm  intersects  speed  circle 
at  apparent  wind  velocity  in  knots.    Clamp  discs  together ;  unclamp  radial  arm. 

(3)  Eotate  radial  arm  until  range  scale  line  intersects  deflection  circle  at  angle 
to  line  of  fire  at  which  wind  is  blowing. 

(4)  Eead  across  by  range  scale  line  to  correction  circle,  and  note  result;  the 
desired  error  in  knots. 

(5)  Determine  sign  of  error  by  glance  at  plotting  board. 

(d)  To  Determine  Errors  Due  to  Motion  of  Gun  (or  Target)  at  Given  Speed 
and  Angle  with  Line  of  Fire. 

Error  in  Range. —  (Use  black  curve,  Column  15.) 

(1)  Eotate  radial  arm  until  target  and  gun  range  curve  intersects  range  scale 
on  radial  arm  at  given  range,  and  clamp  radial  arm  to  upper  disc. 

(2)  Eotate  lower  disc  until  range  scale  line  on  radial  arm  intersects  speed 
circle  at  speed  of  gun  (or  target)  in  knots.  Clamp  discs  together  and  unclamp 
radial  arm. 

(3)  Eotate  radial  arm  until  range  scale  line  intersects  range  circle  at  aligle  to 
line  of  fire  made  by  course  of  gun  (or  target). 

(4)  Eead  across  by  range  scale  line  to  correction  circle,  and  note  result;  the 
desired  range  error  in  yards  due  to  motion  of  gun  (or  target). 


/-t,  <7f  x'<7//j3/W£'  <^^/-ye  Xwa^/a'  Civrfr  a 


APPEXDICES  325 

(5)   Determine  sign  of  error  by  glance  at  plotting  board. 
Error  in  Deflection.     (Use  no  curve.) 

(1)  Eotate  upper  disc  until  zero  of  speed  circle  coincides  with  speed  of  gun  (or 
target)  in  knots  on  correction  circle.    Clamp  discs  together. 

(2)  Eotate  radial  arm  until  range  scale  line  intersects  deflection  circle  at  angle 
to  line  of  fire  made  by  course  of  gun  (or  target). 

(3)  Kead  across  by  range  scale  line  to  correction  circle,  and  note  result;  the 
desired  error  in  knots  due  to  motion  of  gun  (or  target) . 

(4)  Determine  sign  of  error  by  glance  at  plotting  board. 

NoTKS. — 1.  In  all  the  above  described  operations,  the  position  of  the  decimal  point 
at  each  step  must  be  fixed  bj'  the  operator's  general  knowledge  of  the  conditions  in  each 
case. 

2.  By  "  angle  "  is  meant  that  angle  less  than  90°  which  the  course  of  the  firing  ship, 
direction  of  apparent  wind,  or  course  of  the  target  ship  makes  with  the  line  of  fire. 

3.  The  corrections  to  be  applied  to  the  sights  are  numerically  equal  to  the 
determined  errors,  but  of  opposite  sign. 

4.  Variations  from  standard  in  powder  temperature  and  density  of  atmosphere 
cause  errors  in  range  only;  none  in  deflection. 

26.  In  this  paragraph  is  given  the  mathematical  demonstration  that  the 
quantites  given  in  the  tables  for  the  value  of  the  density  of  the  atmosphere  and  of 
the  multipliers  for  Column  12  of  the  range  tables  are  theoretically  slightly  in  error. 

(a)  The  table  of  multipliers  for  Column  12  of  the  range  tables  to  be  found  on 
pages  7  and  8  of  Bureau  of  Ordnance  Pamphlet  No.  500,  on  the  ]\Iethod  of  Computing 
Eauge  Tables  (and  in  Table  IV  of  the  Eange  and  Ballistic  Tables,  edition  for  use 
at  the  Xaval  Academy)  is  based  on  the  standard  table  of  densities  for  given  barometer 
and  thermometer  readings.  This  latter  gives  the  ratio  of  the  density  of  half-saturated 
air  for  a  given  temperature  and  barometric  height  to  the  density  of  half-saturated 
air  at  15°  C.  (59°  F.)  and  750  millimeters  (29.5275")  barometric  height.  The 
values  given  in  the  density  tables  were  computed  from  the  formula : 

g^  1.05498  ^  16     * 


29.4338       .92485 +  .002036^      , 

in  which  H  is  the  barometric  height  in  inches,  t  is  the  temperature  in  degrees  Fahren- 
heit, and  Ft  is  the  vapor  pressure  in  saturated  air  at  t° . 

(b)  Throwing  out  all  constant  multipliers,  this  equation  will  take  the  form 

in  which  x,  y  and  z  are  variables.  Xow  if  we  desire  to  detcrmme  the  values  of  8  for 
different  temperatures  for  any  given  barometric  height,  the  barometric  height 
becomes  a  constant  also  for  the  time  being;  that  is,  x  in  the  above  function  becomes  a 
constant,  and  the  expression  for  the  value  of  8  becomes  an  equation  of  the  first  degree 
involving  only  two  unknown  variables.  Therefore,  all  values  of  S  for  this  particular 
barometric  height  must  lie  on  the  same  straight  line  when  the  curve  is  plotted;  in 
other  words,  the  curve  in  question  must  be  a  straight  line.  The  values  given  in  th'' 
table  do  not  exactly  do  this,  and  are  therefore  in  error  to  the  extent  to  which  they 
deviate  from  this  requirement.  The  errors  are  believed  to  be  due  to  decimal  differ- 
ences in  computation,  and  not  to  be  of  material  magnitude. 

(c)  The  transformations  by  which  the  values  in  the  density  table  are  trans- 
formed into  values  of  the  multipliers  for  Column  12  are  simply  arithmetical  processes, 
and  of  such  a  nature  that  they  do  not  invalidate  the  above  law.  The  same  criticism 
therefore  applies  to  the  values  of  the  multipliers  given  in  the  table. 


APPENDICES  325 

(5)   Determine  sign  of  error  by  glance  at  plotting  board. 
Error  in  Deflection.    (Use  no  curve.) 

(1)  Eotate  upper  disc  until  zero  of  speed  circle  coincides  with  speed  of  gun  (or 
target)  in  knots  on  correction  circle.    Clamp  discs  together. 

(2)  Rotate  radial  arm  until  range  scale  line  intersects  deflection  circle  at  angle 
to  line  of  fire  made  by  course  of  gun  (or  target). 

(3)  Read  across  by  range  scale  line  to  correction  circle,  and  note  result;  the 
desired  error  in  knots  due  to  motion  of  gun  (or  target). 

(4)  Determine  sign  of  error  by  glance  at  plotting  board. 

Notes. — 1.  In  all  the  atove  described  operations,  the  position  of  the  decimal  point 
at  each  step  must  be  fixed  by  the  operator's  general  knowledge  of  the  conditions  in  each 
case. 

2.  By  "  angle  "  is  meant  that  angle  less  than  90°  which  the  course  of  the  firing  ship, 
direction  of  apparent  wind,  or  course  of  the  target  ship  makes  with  the  line  of  fire. 

3.  The  corrections  to  be  applied  to  the  sights  are  numerically  equal  to  the 
determined  errors,  but  of  opposite  sign. 

4.  Variations  from  standard  in  powder  temperature  and  density  of  atmosphere 
cause  errors  in  range  only;  none  in  deflection. 

26.  In  this  paragraph  is  given  the  mathematical  demonstration  that  the 
quantites  given  in  the  tables  for  the  value  of  the  density  of  the  atmosphere  and  of 
the  multipliers  for  Column  12  of  the  range  tables  are  theoretically  slightly  in  error. 

(a)  The  table  of  multipliers  for  Column  12  of  the  range  tables  to  be  found  on 
pages  7  and  8  of  Bureau  of  Ordnance  Pamphlet  No.  500,  on  the  ]\Iethod  of  Computing 
Range  Tables  (and  in  Table  IV  of  the  Range  and  Ballistic  Tables,  edition  for  use 
at  the  Naval  Academy)  is  based  on  the  standard  table  of  densities  for  given  barometer 
and  thermometer  readings.  This  latter  gives  the  ratio  of  the  density  of  half-saturated 
air  for  a  given  temperature  and  barometric  height  to  the  density  of  half-saturated 
air  at  15°  C.  (59°  F.)  and  750  millimeters  (29.5275")  barometric  height.  The 
values  given  in  the  density  tables  were  computed  from  the  formula : 

g^  1.05498  16     ' 


29.4338       .924854-.002036i      . 

in  which  H  is  the  barometric  height  in  inches,  t  is  the  temperature  in  degrees  Fahren- 
heit, and  Ft  is  the  vapor  pressure  in  saturated  air  at  t° . 

(b)   Throwing  out  all  constant  multipliers,  this  equation  will  take  the  form 


«=^(^!) 


in  which  x,  y  and  z  are  varial)les.  Now  if  we  desire  to  determine  the  values  of  S  for 
different  temperatures  for  any  given  barometric  height,  the  barometric  height 
becomes  a  constant  also  for  the  time  being;  that  is,  x  in  the  above  function  becomes  a 
constant,  and  the  expression  for  the  value  of  8  becomes  an  equation  of  the  first  degree 
involving  only  two  unknown  variables.  Therefore,  all  values  of  8  for  this  particular 
barometric  height  must  lie  on  the  same  straight  line  when  the  curve  is  plotted;  in 
other  words,  the  curve  in  question  must  be  a  straight  line.  The  values  given  in  th" 
table  do  not  exactly  do  this,  and  are  therefore  in  error  to  the  extent  to  which  they 
deviate  from  this  requirement.  The  errors  are  believed  to  be  due  to  decimal  differ- 
ences in  computation,  and  not  to  be  of  material  magnitude. 

(c)  The  transformations  by  which  the  values  in  the  density  table  are  trans- 
formed into  values  of  the  multipliers  for  Column  12  are  simply  arithmetical  processes, 
and  of  such  a  nature  that  they  do  not  invalidate  the  above  law.  The  same  criticism 
therefore  applies  to  the  values  of  the  multipliers  given  in  the  table. 


APPENDIX  C. 
ARBITRARY  DEFLECTION  SCALES  FOR  GUN  SIGHTS. 

INTRODUCTORY. 

1.  In  many  cases,  notably  in  turret  sights,  the  system  of  marking  the  deflection 
scales  of  sights  in  "knots,"  as  described  in  this  text  book,  is  no  longer  carried  out; 
these  scales  being  marked  in  arbitrary  divisions  instead,  the  manner  of  constructing 
and  using  which  scales  will  now  be  explained. 

2.  The  method  of  controlling  deflection  by  means  of  "  deflection  boards  "  and 
"  arbitrary  scales  '"'  was  devised  for  the  purpose  of  relieving  the  sight  setters  of  the 
responsibility  of  keeping  the  deflection  pointer  on  a  designated  deflection  curve.  The 
principle  upon  which  the  method  is  based  is  in  no  way  different  from  the  standard 
method  of  controlling  deflection  by  means  of  knot  curves.  It  differs  in  the  method 
of  application,  in  that  one  curve  sheet  upon  which  the  knot  curves  are  drawn  per- 
forms the  functions  of  the  curve  drums  formerly  fitted  upon  each  individual  sight. 
Many  of  the  sights  still  in  service  are  adapted  for  the  use  of  either  method  of  deflec- 
tion control,  and  it  will  be  seen  by  trying  both  methods  that  they  give  the  same 
results,  regardless  of  which  one  is  used. 

3.  The  method  of  bringing  the  point  of  impact  on  the  target  in  deflection  in  no 
way  differs  from  that  of  bringing  the  point  of  impact  on  the  target  in  range,  except 
that  deflection  correction  controls  the  angle  of  the  sight  with  respect  to  the  axis  of 
the  gun  in  the  horizontal  plane,  while  range  correction  controls  it  in  the  vertical 
plane.  If  the  point  of  impact  be  short  of  the  target,  or,  in  other  words,  too  low,  the 
sight  is  raised :  if  the  point  of  impact  is  to  the  left  in  deflection,  the  rear  end  of  the 
telescopic  sight  is  moved  to  the  right,  and  vice  versa.  In  either  case  it  is  the  angle 
between  the  axis  of  the  telescope  and  the  axis  of  the  gun  that  is  changed,  for  range 
in  the  vertical  plane,  and  for  deflection  in  the  horizontal  plane. 

4.  To  arrive  at  a  clear  understanding  of  the  principle  of  deflection,  it  should  be 
comprehended  that  all  deflection  measurements  can  be  reduced  to  angular  measure- 
ments. If  the  horizontal  angle  between  the  axis  of  the  gun  and  the  line  of  sight  be 
the  same  for  all  the  guns  of  the  same  caliber  firing,  then  the  corresponding  deflec- 
tion, whether  measured  in  knots  or  in  yards,  will  also  be  the  same  for  all  those  guns. 
It  is  thus  seen  that  the  sights  for  all  types  can  be  so  constructed  that  the  unit  of 
measurement  for  deflection  is  an  angle. 

PRINCIPLE  OF  ARBITRARY  SCALES. 

5.  In  the  method  of  controlling  deflection  by  the  use  of  "  deflection  boards  " 
and  "  arbitrary  scales,"  the  unit  of  measurement,  that  is,  the  angle  corresponding 
to  one  division  of  the  scale,  is  the  angle  that  is  subtended  by  one-half  of  a  chord  of 
0.2  of  an  inch  at  100"  radius ;  that  is,  it  is  the  angle  whose  tangent  is  .001.  By  using 
this  unit  of  measurement,  the  divisions  on  the  arbitrary  scale  {G,  Plate  III),  are  all 
equal  to  0.1  of  an  inch  on  all  deflection  boards  for  all  sights  for  all  guns,  and  all 
deflection  boards  are  therefore  uniform  in  construction.  The  arbitrary  scale  fitted  to 
each  sight  is  graduated  so  that  one  division  of  the  sight  scale  corresponds  to  this 


APPENDICES  327 

standard  angle,  whatever  the  value  of  the  sight  radius,  and  the  actual  magnitude  of 
each  such  division  in  fractions  of  an  inch  therefore  depends  upon  the  value  of  the 
sight  radius,  and  is  determined  from  it  by  proportion,  as  follows: 

X         0.1  ,  / 

I         100  '  1000 

where  x  (in  fractions  of  an  inch)  is  the  magnitude  of  the  arbitrary  division,  and  I  is 
the  sight  radius  in  inches.  These  arbitrary  scales,  when  once  graduated,  become 
permanent,  regardless  of  any  change  in  initial  velocity  or  other  modifications  affect- 
ing the  trajectory.  The  necessary  corrections  to  provide  for  a  change  in  initial 
velocity,  for  instance,  would  be  made  on  the  curve  sheet  {J,  Plate  III),  and  expensive 
and  troublesome  modifications  in  the  manufactured  scales  on  the  sights  would  there- 
fore be  unnecessary.  As  the  above-mentioned  curve  sheets  are  made  on  drawing 
paper,  quickly  and  at  small  cost,  it  will  be  seen  that  changes  in  the  ballistics  of  the 
guns  could  be  made  without  great  expense  or  delay  in  the  supply  of  the  necessary 
means  for  deflection  control. 

6.  In  the  triangle  under  consideration,  the  "  side  opposite  "  to  the  angle  adopted 
as  the  standard  angular  unit  of  deflection,  that  is,  the  angle  whose  tangent  is  .001, 
is  sometimes  known  as  a  "mill,"  because  the  side  opposite  is  always  one  one-thousandth 
part  of  the  side  adjacent.  In  this  case  it  is  therefore  the  angle  that  corresponds  to  a 
deflection  of  1  yard  at  1000  yards  range,  and  to  a  deflection  of  10  yards  at  10,000 
yards  range,  etc. 

GENERAL  DESCRIPTION  OF  THE  SIGHT  DEFLECTION  BOARD. 

7.  The  "  sight  deflection  board,"  as  shown  on  Plate  III,  as  furnished  to  ships, 
is  simply  a  means  of  mechanically  turning  a  determined  deflection  in  knots  into 
the  units  of  the  arbitrary  scale,  and  at  the  same  time  applying  the  drift  correc- 
tion for  the  given  range.  It  consists  of  a  wood  or  aluminum  board.  A,  about  20" 
square.  On  each  side  is  a  rack,  B,  which  is  secured  by  wing  nuts,  C.  Across  the  top, 
and  also  held  by  the  wing  nuts  C,  is  a  metal  strip,  D,  which  carries  the  sliding 
pointer,  E.  The  scale  of  arbitrary  divisions,  G,  slides  up  and  down  the  board  parallel 
to  itself  upon  the  racks,  B,  as  guides.  A  pinion  on  each  end  of  the  shaft,  F,  runs 
upon  the  racks,  B,  and  prevents  canting  of  the  scale,  G.  The  sliding  pointer,  H,  is 
carried  upon  the  scale,  G,  for  use  in  keeping  track  of  the  divisions  of  the  scale  used. 
The  curve  sheet,  J ,  is  cut  to  fit  under  the  racks,  B,  where  it  is  held  from  slipping, 
after  being  properly  adjusted,  by  the  wing  nuts,  C.  In  placing  the  sheet  on  the 
board,  it  must  be  so  adjusted  that  the  reference  line,  XX,  will  always  be  under  the 
"  50  "  mark  of  the  scale  G  as  the  latter  is  run  up  and  down  from  top  to  bottom  of  the 
board.  (It  will  be  noted  on  the  plate  that  the  line  AM',  which  sliould  intersect  the 
50  curve  at  zero  yards  range,  is  slightly  to  the  right  of  that  curve  at  the  1000-yard 
range  mark  at  the  top  of  the  curve  sheet,  which  is  of  course  as  it  should  be.  The 
slight  divergence  of  the  50  mark  of  the  scale  G  from  the  line  XX  that  is  noticeable 
on  the  plate  is  undoubtedly  due  to  parallax  in  taking  the  photograph,  the  camera 
apparently  not  having  been  set  up  directly  in  front  of  that  point.) 

8.  The  legend  on  the  curve  sheet  shows  for  what  sights,  for  what  caliber  of  gun, 
and  for  what  initial  velocity  it  is  to  be  used ;  and  also  indicates,  for  the  information 
of  the  spotters  and  sight  setters,  the  value  in  knots  at  some  given  range  to  which  the 
divisions  on  the  arbitrary  scale  correspond. 


338 


APPEXDKT.S 


METHOD  OF  USE. 

9.  The  deflection  board  is  designed  primarily  for  use  in  the  plotting  room,  but 
it  can  be  used  at  any  other  point  that  may  be  desired,  such  as  the  spotter's  top  or  in 
the  turrets. 

10.  When  about  to  open  fire,  the  knot  curve  to  be  used  should  be  determined  by 
computation  (or  by  the  use  of  the  gun  error  computer)  in  the  same  manner  as  has 
been  explained  for  the  deflection  sight  scale  marked  in  knots;  but  this  would  no 
longer  be  sent  out  to  the  guns  as  the  setting  of  the  sights  in  deflection.  Instead,  the 
pointer  E  is  placed  at  the  top  to  indicate  the  curve  to  be  used  (the  45-knot  curve  on 
Plate  III).  The  scale  G  is  then  run  down  the  board  to  correspond  to  the  range  to  be 
used  (14,000  yards  on  the  plate).  The  pointer  H  is  then  run  along  the  scale  G  until 
it  is  over  the  proper  curve  on  the  sheet  (45  knots),  and  the  reading  under  the  same 
pointer  on  the  scale  G  will  then  be  the  number  of  divisions  of  the  arbitrary  sight 
scale  at  which  the  sights  should  be  set  to  give  the  desired  deflection  (40  divisions  on 
the  plate).  As  the  curves  on  the  sheet  are  the  drift  curves  for  the  gun,  the  sight 
setting  in  arbitrary  divisions  of  the  scale  thus  found  will  of  course  include  the  drift 
correction. 

11.  As  the  range  varies  during  the  firing,  the  scale  G  is  moved  up  and  down  to 
follow  it,  and  the  pointer  H  is  moved  to  the  right  or  left  to  keep  it  over  the  proper 
curve  on  the  sheet  (45  on  the  plate) .  The  pointer  H  will  then  always  indicate  on  the 
scale  G  the  proper  sight  setting  in  deflection  in  the  markings  of  the  arbitrary  scale. 

12.  In  case  the  spotter's  corrections  indicate  the  use  of  a  new  curve  at  any  time, 
the  pointer  E  is  shifted  to  that  curve,  and  the  new  readings  for  the  arbitrary  scale  are 
read  off  from  the  scale  G  by  the  pointer  H  (which  is  now  following  the  new  curve) 
and  sent  to  the  sight  setters. 

13.  By  this  process  the  sight  setters  are  relieved  of  all  responsibility  in  regard 
to  the  deflection  setting  other  than  that  of  setting  the  sight  for  the  scale  readings 
which  they  receive  from  time  to  time  from  the  deflection  operator,  and  it  is  no  longer 
necessary  for  them  to  be  continually  following  a  drift  curve  on  the  sight  drum  as  the 
range  increases  or  decreases. 

CONTROL  OF  DEFLECTION  WITHOUT  THE  USE  OF  KNOT  CURVES. 

14.  If  preferred,  or  if  no  deflection  board  be  at  hand,  a  table  may  be  made  up 
showing  the  value  in  knots  of  one  or  more  divisions  of  the  arbitrary  dtflection  scale  at 
different  ranges,  from  which  the  spotter  can  estimate  first  his  initial  sight  setting  in 
deflection,  and  afterwards  any  changes  that  he  may  desire  to  make.  So  far  as  the 
deflection  in  yards  is  concerned,  all  that  is  necessary  is  a  knowledge  of  the  fact  that 
for  all  guns,  at  all  ranges,  with  all  arbitrary  deflection  scale  sights,  one  division  of 
the  arbitrary  scale  causes  or  corrects  a  deflection  in  yards  equal  to  one  one-thousandth 
of  the  range;  while  for  the  transformation  from  "knots"  to  "arbitrary  divisions" 
and  the  reverse,  a  table  can  be  used  similar  to : 


Range  in  Yards. 

Deflection  in — 

Knots. 

Divisions  of  Arbitrary  Scale. 

5000 

loono 

15000 
20000 

5 
5 
5 
5 

3.5 
4.0 
4+ 
4.5 

APPENDICES  329 

It  will  be  seen  from  the  above  table  that,  for  the  gun  in  question,  the  variation 
between  knots  and  divisions  is  so  slight  for  all  probable  battle  ranges  that  no  great 
error  would  result  from  assuming  that  5  knots  always  equal  4  divisions  at  all  such 
ranges. 

15.  The  method  of  control  described  above  involving  the  use  of  the  board  gives 
greater  accuracy  than  the  one  using  curve  drums  on  the  sights,  as  the  deflection  board 
permits  the  curve  sheets  to  be  made  on  a  larger  scale.  It  is  not  necessary,  however, 
to  continue  the  use  of  the  deflection  board  after  the  initial  data  has  been  obtained  for 
opening  fire.  The  board  may  be  used  to  determine  the  setting  of  the  sights  in  deflec- 
tion by  the  arbitrary  scale  for  firing  the  first  shot;  and  after  that  the  spotter  can 
indicate  the  deflection  changes  in  terms  of  the  arbitrary  scale,  providing  he  knows 
approximately  the  value  of  the  arbitrary  divisions  in  knots  or  yards  at  the  target,  for 
the  approximate  range  at  which  the  firing  is  being  conducted  (in  yards,  this  is  one 
one-thousandth  of  the  range  in  yards,  as  already  seen)  ;  and,  after  shots  are  seen  to  be 
hitting  at  the  proper  point,  they  can  then  be  held  at  that  point  by  giving  the  spotter's 
corrections  in  terms  of  the  arbitrary  scale,  as  soon  as  the  point  of  impact  appears  to 
creep  to  the  right  or  to  the  left,  and  before  it  can  creep  ofi:  the  target. 

PROPOSED  USE  OF  ARBITRARY  DIVISIONS  FOR  RANGE  SCALES. 

16.  The  advisability  of  using  the  "  arbitrary  scale  "  method  for  ranges  as  well 
as  for  deflection  (replacing  "yards"  in  range)  is  under  consideration,  and  should 
it  be  done,  the  same  division  (one  mill)  would  be  used.  This,  with  a  sight  radius  of 
100  inches,  would  give  clearly  read  divisions  on  the  sight  scale  corresponding  to 
angular  differences  of  two  minutes  in  arc  in  elevation. 

17.  The  use  of  the  arbitrary  scale  in  range  would  avoid  any  changes  in  sight 
graduations  due  to  possible  changes  in  initial  velocity,  as  the  divisions  of  the  arbitrary 
scale  could  be  taken  from  a  range  chart  drawn  on  paper.  This  chart  could  be  so 
drawn  as  to  permit  of  compensation  for  slight  changes  in  initial  velocity  due  to 
variations  in  the  powder,  in  powder  temperature,  etc.  Also,  as  all  divisions  of  the 
sight  scale  would  be  of  equal  magnitude,  this  system  would  lend  itself  readily  to  the 
employment  of  some  step  by  step  mechanical  means  of  setting  the  sights  directly 
from  the  plotting  room  or  elsewhere,  without  the  interposition  of  any  person  as  a 
sight  setter. 


B      C 


he 
to 
ids 
es. 
es, 

m, 
.on 


:tion  board. 


PLATE  III.— DEFLECTION  BOARD. 


ATMOSPHERIC  DENSITY  TABLES, 

BEING  REPRINTS  OF 

TABLE  III  AND  TABLE  IV, 

FROM   THE 

EANGE  AND  BALLISTIC  TABLES,  1914, 

PRINTED  TO  ACCOMPANY  THIS 

TEXT  BOOK  OF  EXTERIOR  BALLISTICS. 

NOTES. 

1.  By  the  nsc  of  tliese  two  tables,  especially  of  Table  IV,  in  conjunction  with  the 
range  table  for  any  particular  gun,  may  be  solved  all  practical  problems  relating  to 
the  use  of  the  range  table  in  controlling  the  fire  of  that  particular  gun,  by  the  methods 
explained  in  Chapter  17  of  this  text  book,  on  the  practical  use  of  the  range  tables. 
These  two  tables  are  all  that  is  necessary,  also,  in  conjunction  with  the  range  tables, 
for  the  solution  of  calibration  problems,  as  given  in  Chapters  18  and  19. 

3.  If  it  be  desired  to  solve  general  ballistic  problems  for  any  particular  gun, 
however,  it  will  be  necessary  to  have  at  hand  the  other  tables  contained  in  the  edition 
of  Range  and  Ballistic  Tables  printed  to  accompany  this  text  book. 


ATMOSPHERIC  DENSITY  TABLES, 

BEING  EEPRINTS  OF 

TABLE  III  AND  TABLE  IV, 

FROM   THE 

EANGE  AXD  BALLISTIC  TABLES,  1914, 

PRINTED  TO  ACCOMPANY  THIS 

TEXT  BOOK  OF  EXTERIOE  BALLISTICS. 

NOTES. 

1.  By  the  iisc  of  tliese  two  tables,  especially  of  Table  IV,  in  conjunction  with  the 
range  table  for  any  particular  gun,  may  be  solved  all  practical  problems  relating  to 
the  use  of  the  range  table  in  controlling  the  fire  of  that  particular  gun,  by  the  methods 
explained  in  Chapter  17  of  this  text  book,  on  the  practical  use  of  the  range  tables. 
These  two  tables  are  all  that  is  necessary,  also,  in  conjunction  with  the  range  tables, 
for  the  solution  of  calibration  problems,  as  given  in  Chapters  18  and  19. 

2.  If  it  be  desired  to  solve  general  ballistic  problems  for  any  particular  gun, 
however,  it  will  be  necessary  to  have  at  hand  the  other  tables  contained  in  the  edition 
of  Kange  and  Ballistic  Tables  printed  to  accompany  this  text  book. 


INTRODUCTION  TO  TABLE  III 

1.  Tliere  are  given  in  this  table  values  of  8,  the  ratio  of  the  density  of  half-saturated  air  for  a  given  temperature  and  barometric  height  to  the  density  of 
half-saturated  air  for  15°  C.  (59°  F.),  and  750  mm.  (29.5275  inches)  barometric  height.  These  values  are  computed  by  the  formula  8  = 
which  H  is  barometric  height  in  inches,  t  is  temperature  Fahrenheit,  and  Ft  is  the  vapor  pressure  in  saturated  air  at  t°. 


H- 


r^t 


29.4338  ^  .92485 +  .0020361' 


TABLE  III 


t[ 

28  in. 

29  in. 

30  in. 

31  in. 

tl 

28  in. 

29  in. 

30  in. 

31  in. 

tf 

23  in. 

29  in. 

30  in. 

31  in. 

U 

28  in. 

29  in. 

30  in. 

31  in. 

0 

1.073 

1.112 

1.150 

1.188 

25 

1.017 

1.053 

1.088 

1.125 

50 

.966 

1.000 

1.035 

1.069 

75 

.917 

.950 

.982 

1.016 

1 

1.071 

1.110 

1.148 

1.186 

26 

1.015 

1.05] 

1.086 

1.123 

51 

.964 

.998 

1.033 

1.067 

76 

.915 

.948 

.980 

1.014 

2 

1.069 

1.108 

1.146 

1.184 

27 

1.013 

1.049 

1.084 

1.121 

52 

.962 

.996 

1.031 

1.065 

77 

.913 

.946 

.978 

1.012 

3 

1.066 

1.105 

1.143 

1.181 

28 

1.011 

1.047 

1.082 

1.119 

53 

.960 

.994 

1.029 

1.063 

78 

.912 

.945 

.977 

1.010 

4 

1.064 

1.103 

1.140 

1.178 

29 

1.009 

1.045 

1.080 

1.117 

54 

.958 

.992 

1.027 

1.061 

79 

.910 

.943 

.975 

1.008 

5 

1.062 

1.100 

1.137 

1.175 

30 

1.007 

1.043 

1.078 

1.115 

55 

.956 

.990 

1.024 

1.058 

80 

.908 

.941 

.973 

1.006 

6 

1.060 

1.098 

1.135 

1.173 

31 

1.005 

1.041 

1.076 

1.113 

56 

.954 

.988 

1.022 

1.056 

81 

.906 

.939 

.971 

1.004 

7 

1.057 

1.095 

1.132 

1.170 

32 

1.003 

1.039 

1.074 

1.111 

57 

.952 

.986 

1.020 

1.054 

82 

.904 

.937 

.969 

1.002 

8 

1.055 

1.093 

1.130 

1.168 

33 

1.000 

1.036 

1.071 

1.108 

68 

.950 

.984 

1.018 

1.052 

83 

.903 

.935 

.967 

1.000 

9 

1.052 

1.090 

1.127 

1.165 

34 

.998 

1.034 

1.069 

1.105 

59 

.948 

.982 

1.016 

1.050 

84 

.901 

.933 

.965 

.998 

10 

1.050 

1.088 

1.125 

1.163 

35 

.996 

1.031 

1.066 

1.102 

CO 

.946 

.980 

1.014 

1.048 

85 

.899 

.931 

.903 

.995 

11 

1.048 

1.08G 

1.123 

1.161 

36 

.994 

1.029 

1.064 

1.100 

61 

.944 

.978 

1.012 

1.046 

86 

.897 

.929 

.961 

.993 

12 

1.046 

1.084 

1.121 

1.169 

37 

.992 

1.027 

1.062 

1.098 

62 

.942 

.970 

1.010 

1.044 

87 

.895 

.927 

.959 

.991 

13 

1.043 

1.081 

1.118 

1.156 

38 

.990 

1.025 

1.060 

1.09G 

63 

.941 

.974 

1.008 

1.042 

88 

.893 

.925 

.957 

.989 

14 

1.041 

1.079 

1.116 

1.153 

39 

.988 

1.023 

1.058 

1.094 

64 

.939 

.972 

1.006 

1.040 

89 

.891 

.923 

.956 

.987 

IB 

1.039 

1.077 

1.113 

1.150 

40 

.986 

1.021 

1.056 

1.092 

05 

.937 

.970 

1.003 

1.037 

90 

.889 

.921 

.953 

.985 

10 

1.037 

1.074 

1.110 

1.147 

41 

.984 

1.019 

1.054 

1.090 

06 

.935 

.908 

1.001 

1.035 

91 

.887 

.919 

.951 

.983 

17 

1.035 

1.072 

1.108 

1.145 

42 

.982 

1.017 

1.052 

1.088 

67 

.933 

.906 

.999 

1.033 

92 

.885 

.917 

.949 

.981 

18 

1.032 

1.0C9 

1.105 

1.142 

43 

.980 

1.015 

1.050 

1.085 

68 

.931 

.964 

.997 

1.031 

93 

.884 

.910 

.947 

.979 

19 

1.030 

1.067 

1.103 

1.140 

44 

.978 

1.013 

1.048 

1.083 

69 

.929 

.962 

.995 

1.029 

94 

.882 

.914 

.945 

.977 

20 

1.028 

1.065 

1.101 

1.138 

45 

.976 

1.011 

1.046 

1.081 

70 

.927 

.900 

.993 

1.027 

95 

.880 

.912 

.943 

.975 

21 

1.026 

1.063 

1.099 

1.136 

46 

.974 

1.008 

1.043 

1.078 

71 

.925 

.968 

.991 

1.025 

96 

.878 

.910 

.941 

.973 

22 

1.024 

1.061 

1.097 

1.134 

47 

.972 

1.006 

1.041 

1.076 

72 

.923 

.956 

.989 

1.023 

97 

.876 

.908 

.939 

.971 

23 

1.021 

1.068 

1.094 

1.131 

48 

.970 

1.004 

1.039 

1.073 

73 

.921 

.954 

.987 

1.021 

98 

.874 

.906 

.937 

.969 

24 

1.019 

1.056 

1.091 

1.128 

49 

.968 

1.002 

1.037 

1.071 

74 

.919 

.952 

.985 

1.019 

99 

.872 

.904 

.935 

.967 

2B 

1.017 

1.053 

1.088 

1.125 

50 

.966 

1.000 

1.035 

1.009 

75 

.917 

.950 

.982 

1.010 

100 

.870 

.902 

.933 

.965 

INTRODUCTION  TO  TABLE  IV 

1.  Tliis  table  is  to  replace  Table  III  for  handy  use  in  a  certain  specific  case.  Column  12  of  the  Range  Tables  gives  the  change  in  ranee  resulting  from  a  raria- 
tion  of  ±  10%  in  the  density  of  the  atmosphere  from  the  standard.  To  use  this  data,  by  the  use  of  Table  111,  it  is  necessary  to  determine  from  Table  111  the 
percentage  variation  in  density,  and  then  apply  this  to  the  data  in  column  12.  To  use  this  table,  however,  take  from  it  the  multiplier  corresponding  to  the  given 
atmospheric  conditions  and  from  column  12  of  the  range  tables  the  number  of  yards  change  in  range  caused  by  a  variation  in  density  of  ±  10%,  multiply  both 
together,  and  the  product,  with  the  sign  of  the  multiplier,  will  be  the  variation  in  range  due  to  the  variation  from  standard  of  the  existing  atmospheric  con- 
ditions. 

TABLE  IV.     MULTIPLIERS  FOR  COLUMN  12  OF  RANGE  TABLES 
Arguments,  Temperature  and  Barometric  Pressure 


tl 

28  in. 

20  in. 

30  in. 

31  in. 

tl 

23  in. 

29  in. 

30  in. 

31  in. 

t( 

28  in. 

29  in. 

30  in. 

31  in. 

U 

23  in. 

29  in. 

30  in. 

31  in. 

0 

—.73 

—1.12 

—1.50 

—1.88 

25 

—  .17 

—  .53 

—  .88 

—1.25 

60 

.34 

.00 

—.35 

—.69 

75 

.83 

.50 

.18 

—  .16 

1 

—.71 

—1.10 

-1.48 

—1.80 

26 

—.15 

—  .61 

-.80 

—1.23 

51 

.36 

.02 

—.33 

—  .67 

76 

.85 

.52 

.20 

—  .14 

2 

—.69 

—1.08 

—1.46 

—1.84 

27 

—.13 

-.49 

—  .84 

—1.21 

62 

.38 

.04 

—.31 

—  .65 

77 

.87 

.54 

.22 

—  .12 

3 

—.66 

—1.05 

—1.43 

—1.81 

28 

—.11 

—.47 

—  .82 

—1.19 

53 

.40 

.06 

—.29 

—  .63 

78 

.88 

.55 

.23 

—.10 

4 

—.64 

—1.03 

—1.40 

—1.78 

29 

—.09 

—  .45 

—.80 

—1.17 

64 

.42 

.08 

—  .27 

-.01 

79 

.90 

.57 

.25 

—  .08 

6 

—.62 

—1.00 

—1.37 

—1.76 

30 

—  .07 

—  .43 

—.78 

—1.15 

65 

.44 

.10 

—.24 

—  .58 

80 

.92 

.59 

.27 

—  .06 

6 

—.60 

—  .98 

—1.35 

—1.73 

31 

—  .05 

—  .41 

—  .76 

—1.13 

56 

.46 

.12 

—  .22 

—  .56 

81 

.94 

.61 

.29 

—.04 

7 

—.57 

—  .95 

—1.32 

—1.70 

32 

—.03 

—.39 

—  .74 

—1.11 

67 

.48 

.14 

—  .20 

—  .54 

82 

.96 

.63 

.31 

—.02 

8 

—  .55 

—  .93 

—1.30 

—1.68 

33 

.00 

—  .36 

—.71 

—1.08 

58 

.50 

.16 

—  .IS 

—  .52 

83 

.97 

.65 

.33 

.00 

9 

—.52 

—  .90 

—1.27 

—1.65 

34 

.03 

-.34 

—  .69 

—1.05 

59 

.52 

.18 

-.16 

—  .50 

84 

.99 

.67 

.35 

.02 

10 

—.60 

—  .88 

—1.25 

—1.63 

35 

.04 

—  .31 

—  .66 

—1.02 

60 

.54 

.20 

—  .14 

—  .48 

85 

1.01 

.69 

.37 

.05 

11 

—.48 

—  .86 

—1.23 

—1.61 

3G 

.06 

—  .29 

—  .64 

—1.00 

61 

.66 

.22 

—  .12 

—.46 

86 

1.03 

.71 

.39 

.07 

12 

—.46 

—  .84 

—1.21 

—1.59 

37 

.08 

—  .27 

—  .62 

—  .98 

62 

.68 

.24 

—  .10 

—  .44 

87 

1.05 

.73 

.41 

.09 

13 

—  .43 

—  .81 

—1.18 

—1.56 

38 

.10 

—  .25 

—  .60 

—  .96 

63 

.59 

.26 

—  .08 

—  .42 

88 

1.07 

.75 

.43 

.11 

14 

—  .41 

—  .79 

-1.16 

—1.53 

39 

.12 

—  .23 

—.68 

—  .94 

64 

.61 

.28 

—.06 

—.40 

89 

1.09 

.77 

.45 

.13 

15 

—.39 

—  .77 

—1.13 

—1.50 

40 

.14 

—.21 

—  .56 

—  .92 

65 

.63 

.30 

—.03 

—  .37 

90 

1.11 

.79 

.47 

.15 

16 

—.37 

—  .74 

—1.10 

—1.47 

41 

.16 

—  .19 

—  .54 

—  .90 

66 

.65 

.32 

—  .01 

—  .35 

91 

1.13 

.81 

.49 

.17 

17 

—.35 

—  .72 

—1.08 

—1.45 

42 

.18 

—.17 

—  .52 

—  .88 

67 

.67 

.34 

.01 

—  .33 

92 

1.15 

.83 

.51 

.19 

18 

—.32 

—  .69 

—1.05 

—1.42 

43 

.20 

—  .15 

—.50 

—  .85 

68 

.69 

.36 

.03 

—  .31 

93 

1.16 

.84 

.53 

.21 

19 

—  .30 

-  .67 

—1.03 

—1.40 

44 

.22 

—  .13 

—  .48 

—  .83 

69 

.71 

.38 

.05 

—  .29 

94 

1.18 

.86 

.55 

.23 

20 

—  .28 

—  .66 

-1.01 

—1.38 

45 

.24 

—  .11 

—  .46 

—  .81 

70 

.73 

.40 

.07 

—  .27 

95 

1.20 

.88 

.57 

.25 

21 

—  .26 

—  .63 

—  .99 

—1.36 

46 

.26 

—  .08 

—  .43 

—  .78 

71 

.75 

.42 

.09 

—  .25 

96 

1.22 

.90 

.69 

.27 

22 

—  .24 

—  .61 

—  .97 

—1.34 

47 

.28 

—.06 

—.41 

—  .76 

72 

.77 

.44 

.11 

—  .23 

97 

1.24 

.92 

.61 

.29 

23 

—  .21 

—  .58 

—  .94 

—1.31 

48 

.30 

—  .04 

—  .39 

—  .73 

73 

.79 

.46 

.13 

—  .21 

98 

1.26 

.94 

.63 

.31 

24 

—  .19 

—  .66 

—  .91 

—1.28 

49 

.32 

—  .02 

—  .37 

—  .71 

74 

.81 

.48 

,15 

—  .19 

99 

1.28 

.96 

.65 

.33 

26 

—.17 

—  .53 

—  .88 

—1.25 

50 

.34 

.00 

—  .35 

—  .69 

75 

.83 

.60 

.18 

—.16 

100 

1.30 

.98 

.67 

.35 

LOAfTPtRIOD  ]    \2 
HOME  USE 


^ 


ALL  BOOKS  AAAY  BE  RECALLED  AFTER  7 


DAYS 


^Om  NO.  DD6,  60.,   ./r'''^^E'R^;^,^^;fO^^ 


®s  I 


YD   [6103 


raOBABILITY  TABLE  FOR  USE  WITH  CHAPTERS  20  AND  21. 

a 


Probability  of  a  deviation  less  than  a  in  terms  of  the  ratio 


7 


a 

7  ' 

P. 

a 

7  ' 

P. 

a 

7' 

P. 

a 

7  ' 

P. 

0.1 

.004 

1.1 

.620 

2.1 

.906 

3.1 

.987 

0.2 

.127 

1.2 

.662 

2.  2 

.921 

3.2 

.990 

0.3 

.189 

1.3 

.700 

2.3 

.934 

3.3 

.992 

0.4 

.250 

1.4 

.735 

2.4 

.945 

3.4 

.994 

0.5 

.310 

1.5 

.768 

2.5 

.954 

3.5 

.995 

0.6 

.368 

1.6 

.798 

2.6 

.962 

3.6 

.996 

0.7 

.424 

1.7 

.825 

2.7 

.969 

3.7 

.997 

0.8 

.477 

1.8 

.849 

2.8 

.974 

3.8 

.998 

0.9 

.527 

1.9 

.870 

2.9 

.979 

3.9 

.998 

1.0 

.575 

2.0 

.889 

3.0 

.983 

4.0 

.999 

COLUMBIAN  CLASP 

W0HCF.STra.WA55.  flOLYOKC.MiSS.    RQCIV1U£C(»«I, 
iFra«eFl£LJ3.l>lASS.  nABTfOfiacOSN.    WAL'KfttMJU. 

cincmNxri,oHio.   san  franciscq.cal. 


No. 

25 


5i^ 


INTERPOLATION  FORMUL.^. 
For  use  with  Table  II,  Ballistic  Tables. 


(1)  A    =A,+ 

(2)  V   =V,+ 

(3)  Z    =Z,+ 


z-z, 

100 
AV 

100 


AzA  + 


v-v. 


-A,)- 


[(A-A,) 


AV 

z—z 


A.,. 


100  '"^^'J- 


V V 

(4)   A"=A';-^^X 


v-v, 

AV 

+ 


„]. 


A,.,Az,„   ,    ^-^^^  a,,,.+  {A'-A:) 


>\  AzA" 


^ZA' 


AV 


When  V — Ft  =  0,  (4)  becomes 


(5)  a"=a';-\-{A'-a',)^ 


^ZA' 


In  using  the  above  formulie  exercise  great  care  to  use  each  quantity  with  its  proper  sign. 

These  formulae  are  correct  for  working  from  the  next  lower  tabular  value  only;  if  work 
is  to  be  from  the  next  higher  tabular  value  there  must  be  a  general  change  of  signs  in  tlie 
formulae.     Work  from  the  next  lower  tabular  value  unless  directed  to  the  contrary. 


fORM  NO.  DD6,  60.,  .^T^^^^^i^^^^^^^^ 

DtKKbLEY,  CA  94720 


®s 


YD   I&I03 


/•  riL.hJL. 


€) 


PROBABILITY  TABLE 
For  use  with  Chjipters  20  and  21 

and 

INTERPOLATION  FORMULAE 

For  use  with  Table  II,  Ballistic  Tables, 

enclosed 


/ 


\ 


25 


COLUMBIAN  CLASP      ^^ 

WOHCF^rra.lMii.  nOLYOKC.MASS.   ROClVlLLtCOW,       OK 
ClNClHNXn^CHiO.     SAN  FRANC15CQ.C1L. 


1^«     •    m  t.i'i 


5»^ 


